Chicago Lectures in Mathematics Series
Robert J. Zimmer, series editor
J. Peter May, Spencer J. Bloch, Norman R. Lebovits,
William Fulton, and Carlos Kenig, editors
Other Chicago Lectures in Mathematics titles available from the
University of Chicago Press:
Simplicial Objects in Algebraic Topology, by J. Peter May A967)
Fields and Rings, Second Edition, by Irving Kaplansky A969, 1972)
Lie Algebras and Locally Compact Groups, by Irving Kaplansky A971)
Several Complex Variables, by Raghavan Narasimhan A971)
Torsion-Free Modules, by Eben Matlis A973)
Stable Homotopy and Generalised Homology, by J. F. Adams( 1974)
Rings with Involution, by I. N. Herstein A976)
Theory of Unitary Group Representation, by George V. Mackey A976)
Infinite-Dimensional Optimization and Convexity, by Ivar Ekeland and
Thomas TumbullA983)
Navier-Stokes Equations, by Peter Constantin and Ciprian Foias A988)
Essential Results of Functional Analysis, by Robert J, Zimmer A990)
Fuchsian Groups, by Svetlana Katok A992)
Unstable Modules over the Steenrod Algebra and Sullivan's Fixed Point Set
Conjecture, by Lionel Schwartz A994)
Topological Classification of Stratified Spaces, by Shmuel Weinberger
A994)

J. F. Adams
LECTURES ON
EXCEPTIONAL LIE GROUPS
Edited by Zafer Mahmud and Mamoru Mimura
The University of Chicago Press
Chicago and London

J. F. Adams A930-1989) was Lowndean Professor of Astronomy and Geometry at the
University of Cambridge. Zafer Mahmud is associate professor of mathematics at Kuwait
University. Mamoru Mimura is professor of mathematics at Okayama University.
The University of Chicago Press, Chicago 60637
The University of Chicago Press Ltd., London
© 1996 by the University of Chicago
All rights reserved. Published 1996
Printed in the United States of America
04 03 02 01 00 99 98 97 96 12 3 4 5
ISBN 0-226-00526-7 (cloth)
0-226-00527-5 (paper)
Library of Congress Cataloging-in-Publication Data
Adams. J. Frank (John Frank)
Lectures on exceptional Lie groups / J.F. Adams ; edited by Zafer Mahmud and
Mamoru Mimura.
p. cm. —(Chicago lectures in mathematics series)
Includes bibliographical references.
ISBN 0-226-00526-7 (cloth : alk. paper). —ISBN 0-226-00527-5
(ppk. : alk. paper)
1. Lie groups. 1. Mahmud, Zafer. II. Mimura, M. (Mamoru). 1938-
III. Title. IV. Series: Chicago lectures in mathematics.
QA387.A33 1996 96-21456
512'.55—dc20 CIP
^ The paper used in this publication meets the minimum requirements of the American
National Standard for Information Science—Permanence of Paper for Printed Library Materials,
ANSI Z39.48-1984.

Contents/Summary
Summary of Constructions x
Foreword xi
Acknowledgments xii
Introduction xiii
Chapter 1. Definitions, examples and matrix groups 1
• Infinitesimal methods 2
Determination of tangent spaces (Lie algebras)
Adjoint representation
• Representation theory of compact groups 3
Representations, real and symplectic
• Weights and characters 4
« Sketch of classification of compact Lie groups 5
The Weyl group W
Roots
The Stiefel diagram
Weyl chambers, positive and simple roots
Dynkin diagrams
Reductive Lie algebras
Chapter 2. Clifford algebras 13
Definition and universal property
Tensor products, bases for Cl{V)
• Structure maps on Clifford algebras 15
Chapter 3. The Spin groups 17
Definition of Pin{V) and Spin{V)
Maximal tori
Chapter 4. Clifford modules and representations 21
Semi-simplicity of Cl(V) and Cl{V)o
The irreducible representations A, A"*", A"
The weights of A

A is self dual. A"*", A are self dual (resp. dual to each other) for
n even (resp. n odd).
Behaviour of A, A"*", A" under restrictions
Calculations of o-2(A±), \\i\^)
• The theorem of Weyl on R{G) 28
For G compact, connected, simply-connected R{G) is a polynomial
algebra with generators in 1-1 correspondence with the nodes of
the Dynkin diagram.
Chapter 5. Applications of Spin representations 31
• Construction of G2 31
Spin{6) = SU{i), Spin{5) = 5pB).
SpinE) acts transitively on 5^ C A.
Spin{6) acts transitively on 5^ x 5^ c A' x A"*".
Spin{7) acts transitively on {x,y,z) e S^ x S^ x S^, where x ±y.
G2 is the subgroup of Spin[7) fixing a point z G 5^ C A.
G2 is compact, connected, simply connected, of rank 2, dimension 14
acting transitively on [x,y) € S^ x S^ where x ±y.
• Spin{8) and triality 33
Out{Spin{8)) ~ ?3, acting on A+, A", A'.
There is a non-zero trilinear / : A+ (g) A~ ® A' —» C invariant under
5pin(8) and for real representations (of dimension 8), / takes
values in [—1,1].
Chapter 6. The exceptional groups: construction of E% 37
There are pairs of Lie groups H C G with rank H = rank G as follows:
local type of H
Spin{9)
5pnA0)xC/(l)/Z4
5pinA2)x5p(l)/Z2
5pnA6)/Z2
dim
36
46
69
120
L(G)/L(//) as C rep.
A
A+(g)^^-f-A-(g)r^
A+®A'
A+
dim
16
32
64
128
G
F4
?
Ej
Es
dim
52
78
133
248
Construction of a Lie algebra of type Eg
Standard operating procedure
The Killing form
38
40
42

Chapter 7. Construction of a Lie group of type Eg, 45
Step 1. A = L{Spin{l6)) ffi A"*" is a Lie algebra with invariant inner
product and non-singular Killing form.
Step 2. Take the group of Lie algebra automorphisms.
L{Aut(A)) = Der{A).
If A has non-singular Killing form, then Der{A) = A.
Step 3. Take the identity component.
• Real forms of E^ 46
Chapter 8. The construction of Lie groups of type F4, ?, Ey 49
Ej = C°siSUi2)).
Ee = C%,{SU{3)).
Fi = CssCGz), where SU{2) C SU{3) C G2.
• Identification of the subgroups H 50
• Identification of L(G) and L(G)/L{H) 51
• Real forms of Eg, continued 53
Chapter 9. The Dynkin diagrams of F4, Eg, Ey, Eg 55
• Fi 55
• Eg 56
• ? X SC/B)/Z2 56
. Ee X SU{3) 57
Chapter 10. The Weyl group of Eg 59
W{Eg) is the group of all automorphisms of the root system of Eg
(Theorem 10.1).
W{Eg) acts transitively on the figures (Theorem 10.13)
Chapter 11. Representations of Eg, Ej 69
?¦7 has a subgroup SC/(8)/Z2 (Theorem 11.1).
?7 has an irreducible representation of dimension 56 (Corollary 11.2).
Ee has a subgroup SC/C)VZ3-(Theorem 11.4).
? has a 27-dimensional irreducible representation (Corollary 11.5).
Chapter 12. Direct construction of ?7 73

• Construction of L(Ej) and its 56-dimensional representation 74
Step 1. (Theorem 12.1) A = L{SU{8)) ® A'' is the Lie algebra
(= LiEy)) of maps W -^W. Steps 2 and 3 as for Eg.
W admits
a skew-symmetric linear form { , ),
a map i : A —^ W <SW into symmetric tensors,
a map o : W iS> W —^ A.
The action of Ej on W is faithful.
• ? as a group of maps of W (Cartan's construction) 80
W admits an invariant symmetric map W^* —^ C.
? = the identity component of the group of linear maps W -* W
preserving ( , ) and W®'' —> C.
• Real forms of Ey 82
Chapter 13. Direct treatment of ? 85
• Construction of Eg and its 27-dimensional representation 85
A = L{SU{3f) e (Vi ® Vj (g) V3) is the Lie algebra (^ L{Ee))
of maps W —> W.
There is a non-singular inner product ( , ) on A and the Killing form
is 24( , ) (Lemma 13.4).
There is a unique map W ® W —> A such that (a,w ® w') =
{[a,w\,w') (Lemma 13.5).
The action of ? on its 27-dimensional representation is faithful.
• ?6 as a group of maps 91
?6 is the group preserving the products W <SW —^ W*,
Chapter 14. Direct treatment of F4, I 93
F4 has a subgroup of local type SUC) x SU{3) so that the roots of
the first factor are the short roots of F4 (Lemma 14.1).
W{Fi)/W{Spin{8)) S E3 = Out{Spin{8)) (Theorem 14.2).
F4 has an irreducible 26-dimensional representation U (Lemma 14.4).
L{Ee) = L(F4) e U, WIF^ = l + U (Lemma 14.4).
• Structure maps on U 96
U carries the following structure:
a symmetric, bilinear, non-singular, F4-invariant 6 : C/ ® {/ —> C,
a symmetric bilinear F4 invariant epi x : U <SU —f U.

• The algebra structure on U 100
l + U may be constructed as R^eViffiVieVs where {Vi,V2,V^} =
{A + ,A-,A'} (Theorem 14.18).
Chapter 15. The Cayley numbers 105
There is a unique normed algebra of dimension 8 over R
(Theorem 15.1).
If A is a normed real algebra of dimension 8 with 1 and 1, i, j,
ij, ? are orthonormal, H = {l,i,j,ij), then H ffi H —> j4 is an
isomorphism (Theorem 15.8).
O = H ffi H is cJternative (Theorem 15.11).
• Connection between the Cayley numbers and Lie groups 111
By choosing {xo,yo,Zo) ? A"*" ® A" ® A' we can identify A"*", A~,
A' with O (Theorem 15.14).
G2 = AuttiiO) (Theorem 15.16).
Chapter 16. Direct treatment of F4, II: Jordan algebras 113
• Definition and properties of the exceptional Jordan algebra J 113
J carries the following structure:
a product o : J ® J —> J,
a bilinear 6 : J ® J —> R,
a linear ^ : J —> R.
If we identify R^ with the space of diagoneJ matrices, Vi with the
Cayley numbers, then R^ ffi Vi ffi V2 ffl ^ is identified with J so
as to make the structures correspond (Theorem 16.1).
U corresponds to the subspace ? = 0 in J and F4 acts on J so as
to preserve the algebraic structure (Corollary 16.2).
The algebra of polynomial functions on J invariant under Aut{J) =
algebra of polynomial functions on J invariant under i*4
(Theorem 16.6).
F4 = Aut{J).
The subgroup of Aiit{J) fixing ej = A,0,0) is Spin{9).
The subgroup of Aut( J) fixing 61,62,63 is Spin(8) (Theorem 16.7).
• The Cayley projective plane 118
Appendix. Jordan algebras 119
References 121

Summary of Constructions
Construction (possibly) using Eg
G2 = subgroup of Spin{7) fixing z G 5^
F4 = C|JG2),G2 ^ 5pmG) - Spin{16) -^ Eg
Ee = C0,EC/C)),5C/C) -- 5pinA6) - Eg
?•7 = C^,EC/B)), 5C/B) ^ 5pinA6) - Eg
Eg
Direct construction
AutniO)
Aut{J), J = algebra
Auto{W,W',x)
AutoiWA, ),!),!-.w^'-^c
Aut{L{Spin{l6) ® A+))

Foreword
Frank Adams was long intrigued by the exceptional Lie groups, and he wrote
several papers about them ([Adl, Ad2, Ad3] in the bibliography). He worked out
a connected narrative describing their construction, their representations, and
their interconnections and connections with the classical Lie groups in several
lecture series given at Cambridge University. He often contemplated turning his
lecture notes into a book, but unfortunately this was prevented by his untimely
death. At the Memorial Symposium that was held in his honor at the University
of Manchester in August of 1990, Professors Zafer Mahmud and Mamoru Mimura
graciously agreed to edit the several versions of Adams' lecture notes, with the
further aid of the notes taken by Wilson Sutherland and Adams' student, John
Greenlees. Professors Mahmud and Mimura have carefully carried out this task.
Since it was Adams' custom to write out lecture notes in unusual detail, the
resulting volume faithfully reproduces his own words, but it is to be understood
that Mahmud and Mimura were working from lecture notes that Adams himself
would undoubtedly have reworked before publication. Nevertheless, we believe
that the resulting volume is true to Adams' spirit and will be of considerable
value to all mathematicians who wish to learn about those wonderful curiosities,
the exceptional Lie groups. The notes themselves give excellent summaries of the
required background in Lie group theory, details of which may be found in Adams'
own volume Lectures on Lie Groups, which is also available from the University
of Chicago Press.
J. Peter May

Acknowledgments
Zafer Mahmud wishes to thank Professors S. J. Patterson and Larry Smith of
the Mathematics Institute, Gottingen, for very generous hospitality and facilities
during the year 1990-91. Without this support, he could not have participated in
this project.

Introduction
The object of this work is to study the exceptional Lie groups. These may
be regarded as curiosities of Nature though all right-thinking mathematicians are
glad they exist. In particular, some theoretical physicists are beginning to wonder
if the exceptional groups turn up in quantum theory. In algebraic topology, their
homology and cohomology rings provide examples of Hopf algebras to be reckoned
with in formulating general structure theorems. Again, people interested in finite
groups are interested in finite groups related to these Lie groups.
The book should be seen as a sequel to [1]. The aim is to give concrete and
explicit constructions of the exceptional Lie algebras and Lie groups with the most
interesting low dimensional representations. We will not give the general theory
or the most conceptual definitions. Most of what we need can be found in [1].
Before we get to the five exceptional groups, there are other matters we should
mention first: the groups Spin{n) and their representations A (n odd). A* (n
even). This must be preceded by something about Clifford algebras.
To begin with, we have general chapters about Lie groups. This is not because
we are going to need theorems: you can always study an explicit concrete
special case without the aid of general theory. But we need language and standard
constructions, for example: passing from a Lie group to its Lie algebra. More
specifically, we need methods to recognise that the exceptional groups we construct
are not isomorphic to any of the classical groups.
The prerequisites for this book are [1] or a third year undergraduate course
in differential geometry (in a British University), linear algebra (tensor products
and Horn functors etc., such as in [La]) and rudiments of topology. (The US
equivalent prerequisites would be advanced calculus, linear algebra, differential
geometry, basic topology and a first course in abstract algebra.) Thus, where a
map or space ought to be smooth, we hardly ever check that it actually is, leaving
it to the reader to see that the map or space in question is obviously smooth.
Apart from these prerequisites, we supply detailed proofs throughout and make
all constructions ab initio.
The first six chapters are preparatory. In many of our constructions it is
possible to work simultaneously over the reals and complex numbers. If it is not
explicitly stated, the context should make clear the ground field over which we are
working. We construct G2 in Chapter 5, the Lie algebra Eg in Chapter 6 and
the Lie group Eg in Chapter 7. In Chapter 8, F4, ? '^nd Ey are constructed
as the connected components of the centralisers of various subgroups of Eg, and
certain standard representations identified. The Dynkin diagrams are given in
Chapter 9, the Weyl group of Eg discussed in Chapter 10, and properties of the
standard representations of Eg and Ej derived in Chapter 11. Chapters 12, 13,
14, and 16 are largely independent of the rest of the book and give constructions
of F4, Eg, E-r and their representations, not using the group Eg. The idea here
is to construct a vector space equipped with extra structure. This space turns

out to be the representation constructed previously and the groups appear as the
group of automorphisms preserving all the structure in sight. In particular, F4
appears as the automorphism group of an exceptional Jordan algebra (there is
a brief appendix listing simple properties of Jordan algebras). In similar vein,
in Chapter 15 we construct the algebra of Cayley numbers and identify G2 as
the automorphism group. The reader can get an idea of these constructions by
consulting the CONTENTS/SUMMARY. The end of a proof is indicated thus: [].

Chapter 1
Definitions, examples and matrix
groups
We assume familiarity with the basic theory of compact Lie groups and Lie
algebras to be found in [1] (other references are [7], [9]). In this Chapter we establish
the notation that will be used in the sequel.
A Lie group G is a group which is also a smooth manifold such that the maps
G X G —f G, {g,h) >—t gh and G —^ G, g >—^ g~^ are smooth. A homomorphism
0 : G —^ H of Lie groups is a homomorphism of groups which is also a smooth
map, [1, p. 3]. A representation V of G is a representation in the sense of
algebra, [1, p. 23, definitions 3.1], such that the map Gx V -+ V, [g,v) i-» gv is
smooth. Let A = R, C or H be respectively the reals, the complex numbers or
the quaternions and V a finite dimensioned right A-module. Then, by choosing
a basis for V, we get isomorphisms V S A" for some integer n and
End{V) = Endx{V) = Hcmi^{V, V)
= A^„(A) = {nx n matrices with entries in A }.
The general linear group of V,
GL{V) = Aut{V) = {A? End{V) | 3 A"' e End{V) },
is a group, and an open subset of End{V) (= a finite dimensional vector space
over A). This ensures that it is a smooth manifold. The product and inverse map
in GL{V) are smooth, so GL{V) is a Lie group. If the dimension of V over A
is n, we will write GL(n,A) for GL(V). We can choose on V a Hermitian form
( , ) such that
{v,w\) = {v,w)\ for A G A, v,w eV, and
{w,v) = {v,w),
and then consider such subgroups of GL{V) as
{g G GL(n,A) | {gv,gw) = {v,w), Vu,u; G V}.
1

2 Chapter 1
Here {v,w) denotes the conjugate in A, see Example 2 at the end of Chapter 2.
By choosing a basis in V and defining (x, y) — x^y, we see that for all v &V,
{v,v) e R, {v,v) > 0 with = if and only if v = 0.
The subgroups {A S GL{n,A) \ A^A = 7 } are denoted 0(w), U{n), Sp{n)
respectively for A = R, C, H. For A = R or C, we also get subgroups {A?
GL(n,A) I detA = 1 } denoted SL{n,R), SL{n,C) (called the special linear
groups) and
SO{n) = SL{n,R) n 0(n), SU{n) = SL{n,C) n U{n).
All these groups are collectively called classical groups (and are in fact Lie groups),
see [7, pp. 1-24].
Definition. A matrix group is a cJosed subgroup of GL{V) for some V.
Infinitesimal methods.
The set of all tangent vectors at a point P of a manifold X is denoted Xp, and
if X is modelled on a vector space V then Xp will be a vector space isomorphic
to V. For each of the matrix groups we have met, we list the space of tangent
vectors at the identity.
Proposition. Suppose G = 0{n), U{n) or Sp{n). Then B e Gi if and
onJy if B^ + B = 0. For the groups G = SL{n,R),SL{n,C), B sGi if and
only if Tr{B), the trace of B, is zero.
D
Next we note that there is a Lie algebra structure on Gi = L{G) and that L
is a functor from the category of Lie groups and homomorphisms to the category
of Lie algebras and Lie algebra homomorphisms and that a representation of G
gives a representation of L{G).
Theorem 1.2. (i) For any Lie group G, the space Gj of tangent vectors to
G at 1 becomes a Lie aigebra L[G) over R.
(ii) For any matrix group G C GL{V), the bracket [7,E] in L{G) is the
commutator jS — Sj s End(V).
(iii) If 0 : G —> H is a homomorphism of Lie groups, the induced map
01 : Gj —+ fli becomes a homomorphism L{9) : L{G) —^ L[H) of Lie algebras.
(iv) For any representation V^ of G, V becomes a representation of L{G).
(v) For a matrix group G C GL{V) acting on V as given, the action of
L{G) C End(V) is the usual action of End{V) on V.
(vi) If H acts on V and we are given a homomorphism 9 : G -* H of Lie
groups, so that G acts on V, the resulting action of L{G) on V is 'y-v = {6i'y)v
where 0, = L{e) and 7 G L{G).

Definitions, examples and matrix groups 3
A vector space V over A' = R, C or H, A C A', is a representation of a Lie
algebra A if we are given products xv ? V for all x G A, v ? V, A-linear in
X, A'-linear in v and satisfying lx,y]v = x(yv) — y(xv).
G acts on itself by conjugation: for all g G G, we have ig : G —^ G, ig{h) =
ghg~^, which gives (ig)i : Gi -* Gi, so G acts on Gi = L(G). This is the adjoint
representation, Ad : G —^ Aut[L{G)).
References for Theorem 1.2 are [U, pp. 421-432] and [4, Chapter 3, pp. 120-
165],
Definition. A Lie algebra is simple if it is of dimension more than one and
has no proper ideals; a Lie algebra is semisimple if it is has no abelian ideal. A
Lie group is simple (semisimple) if its Lie algebra is simple (semisimple).
For example, SU{n),n > 1, and its Lie algebra are simple; over R or C a Lie
algebra is semisimple if and only if it is the direct sum of simple algebras.
Of course, L[G) only sees a neighbourhood of 1.
Examples, (i) SO{n) is the identity component of 0{n) (which has two
components) and
L[SO{n)) = {real skew sjonmetric nxn matrices } = L[0[n)).
(ii) We note that C/(l) = { z | zz = 1} = 5' C C and the exponential map
R —> 5', ti-* e^", induces an isomorphism of Lie algebras (of R and 5') both
of which are of dimension 1 and have zero brackets. Of course, the groups R and
S' are not isomorphic.
(iii) The quotient map SU{n) -+ PSU(n) = SU{n)/{wI | lu" = 1} gives
an isomorphism of Lie algebras. []
Even so, L(G) does determine G locally, [Ha, p. 71].
Representation theory of compact groups.
(See [1, Chapter 3, pp. 27-78], [3, Chapter 6, pp. 171-213].)
The groups 0{n), U{n), Sp{n), SO{n), SU{n), PSU{n) are compact (whilst
GL{n,A), and SL(n,A), n ^ I, are not).
Most of the representation theory of finite groups goes over to compact groups
with integration a substitute for finite sums. Thus for any real valued continuous
function f on G we can define L^q f{g) and it has the properties expected of
7^ E f{9), [1, p. 33].

Chapter 1
We shall focus on complex representations. For certainly, a representation V
over H is a representation over C. This representation over C, together with
a conjugate linear structure G-map j : V —f V such that j^ = —1, ij = —ji
gives the original H-representation. A representation V over R gives V ®r C
and this carries a conjugate linear structure map j:v(Sz>—>v(Sz such that
j^ = 1 and we can regard V as the +1 eigenspace of j (or the —1 eigenspace).
See [1, pp. 24-33, 44-45].
Weights and characters.
One of the benefits of working over C is that every irreducible representation
of a compact abelian group is 1-dimensional. For example, let T be the torus
d
X 5 . Another way to regard T is that since it is a connected compact abelian
group, exp : L(T) —> T is a homomorphism, [1, p. 15, proof of Theorem 2.19], so
T = L(T)/kerexp. Write F = kerexp; then F is a discrete subgroup of L(T),
called the integer lattice of L(T) and choosing a basis gives L{T) = R'' for some
d, [1, p. 15] and
T ^ L(T)/F S RVZ" S (R/Z)''.
Homomorphisms 9 : T ^f T' are easily described. We need only check, for
a linear map ip : L{T) -> L(T'), that ifi{T) C F', and if so, then 6 = ifi :
L{T)/T —^ L(T')/r'. All continuous homomorphisms arise in this way, cf. [1,
p. 76, Proposition 3.74], and all 1-dimensional representations of T arise from
linear maps ip : L(T) —^ L{U{1)), [1, Corollary 3.75]. Given a representation
V of T, there are linear maps ip : L{T) —> L{U{1)) such that V decomposes
as a direct sum of non-zero sub-representations V^ where L{T) acts on V^ by
t(x) = ifi{r)x (r G L{T),x G V^). These linear maps ip on L(T) are called the
weights of V, and the dimension of V^ the multiplicity of p [1, pp. 75-77]. The
space of tangent vectors to 5' C C exists of course, and it is iR. We will take
F = 27ri Z. (If we insisted on integral coordinates, we would get many useless
factors of 27ri.)
Fig. 1
27ri
=i^3
L(C/(l))=iR

Definitions, examples and matrix groups 5
Suppose y is a representation of G over C. Then the character, Xv ¦ G —^ C,
of V is Xv{9) — '^'"'cid '-V —iV). Characters are class functions, that is,
for all 5,/i e G. Also
Xv®w{9) = Xv{9) + Xw{9)> Xviswig) = Xv{9) ' Xw{9)-
Theorem 1.3. If Xv = Xw ^iien V^W.
See [1, pp. 46-52, especially Proposition 3.37].
Sketch of the classification of compact Lie groups.
This involves constructing the Dynkin diagram of a compact Lie group and
for this we need
Definition. A maximal torus in a compact connected Lie group G is a
subgroup T which is
(i) a torus and (ii) maximai,
that is, if T cT' CG, V a torus, then V = T.
A general reference for maximal tori is [1, pp. 79-100].
Examples 1.4. See [1, examples 4.16-4.20].
(i) In U{n) consider diap(e^"'"',... ,e^""), the diagonal matrices with
diagonal entries e^"'^' for Xj G R. These form a maximal torus in U{n).
(ii) Since C C H, the matrices of (i) are in Sp(n), and they form a torus in
Sp{n). Since C" can be regarded as R^", we get an embedding U(n) —> SOBn)
and we can again take the corresponding matrices, namely
/ cos27rxi — sin27rxi "^
sin27rxi cos27rZi
cos 27rXn — sin 27rxn
y sin27rxn cos27rXn ^
These will form a torus in S0{2n).
(iii) We can embed R^" in R^"^' and thus S0{2n) in S0{2n + 1), where
Ai-* (g jj and so get a corresponding torus in S0{2n + 1).
(iv) In SU{n) we take the matrices of (i) subject to Yl^i = Q to get a
maximal torus.
These tori are in fact all maximal. For example, in U{n) any matrix X
which commutes with aU diagonal matrices must be diagonal, so in T. Thus

6 Chapter 1
T is maximcil among all abelian subgroups, connected or not. The existence of
maximal tori is easy to prove in any Lie group. It is more substantial to prove
that they have the properties one wants; these properties mirror the behaviour of
diagonal matrices in U{n).
Theorem 1.5. Let T C G be a maximal torus of a compact, connected,
Lie group G. Then any g ? G is conjugate to some element of T. That is, there
exists t gT, h G G such that g = hth~^.
See the proof in [1, pp. 89-92]. Q
Corollary 1.6. If V, W are representations of a compact, connected. Lie
group G and Xv\T = Xw\T, then Xv = Xw, so V = W.
Proof. The torus provides at least one representative from each conjugacy
class, see [1, p. 95]. [}
If y is a representation of G, the weights of V are, by definition, the weights
of V\T. The weights (together with their multiplicities) determine Vup to
equivalence.
Corollary 1.7. Given two maximal tori T, T' in a compact, connected. Lie
group G, there exists an inner automorphism of G taking T to T'.
Proof. [1, pp. 92-93]. []
It follows from Corollary 1.7 that any property of G defined by reference to a
maximal torus T is independent of the choice of T.
Example: (Definition.) The rank of a compact, connected. Lie group G
is the dimension of a maximal torus of G. (We wiU usually write i = rank G.)
Definition 1.8. The Weyl group W of a compact, connected. Lie group G
is the group of those automorphisms of a maximal torus which are given by inner
automorphisms of G.
Example. In U{2) conjugation by (° "gj is an element of W and
0 -l\ ( e^'^'^' 0 \ ( 0 1 \ _ / e^'^^ 0
1 oil 0 e^'^'^' j I -1 0 j ~ i 0 e^"^'
More generally, the Weyl groups of our matrix groups are as follows [1, pp. 114-
116]:

Definitions, examples and matrix groups
U{n) and SU{n) : W — Y,n (any pennutation of Xj Xn).
Sp{n) and S0{2n + 1) : W = the group generated by all permutations of
Xi,... ,Xn and all sign changes of Xi.
S0{2n) : W = the group generated by aU permutations of
Xi,... ,Xn and an even number of sign changes of Xi.
D
We now have enough invariants to settle some questions. For example, if we
have a Lie group (such as Eg, see Chapter 8) of dimension 78 and \W\ = 2^ ¦ 3'' ¦ 5
then we know that G ^ 5pF), since \W{Sp{6))\ = 2^ ¦ 6! = 2'" • 3^ • 5.
However, we need to go further.
Suppose G is a Lie group and TcG a torus (not necessarily maximal). Then
G acts on L{G) via the adjoint representation, so T acts on L{G) by restriction
and L{G) ® C splits as a sum of 1-dimensional representations of T, with T
acting trivially on L{T) C L{G). Thus the trivial 1-dimensional representation
occurs at least d = dim T times.
Proposition 1.9. If G is compact, then T is maximal ¦^^=^ the trivial
1-dimensional representation occurs exactly d times.
Proof. [1, p. 83].
From now on suppose T is maximal and put d = L
Definition. The roots of a compact, connected. Lie group G are the weights
of the adjoint representation, excluding 0 (which occurs, by Proposition L9,
? times).
The roots are thus R-linear functions on L{T), that is, elements of L(T)'.
Since the adjoint representation of G is real, the 1-dimensional summands of
L{G) ® C occur in conjugate pairs and the roots occur in pairs ± 6.
Examples. (See [1, pp. 83-89].) We list the weights of LF')® C]T for the
matrix groups G and maximcJ tori we have described in Examples 1.4.
U{n) : weights are ± {Xi — Xj) I < i < j < n, 0 n times.
Proof. First note that L([/(n)) ® C S L{GL{n,C)) S EndiC) since
B I—» —B^ is a conjugate linear structure map and its +1 eigenspace is L{U{n)).
Take the basis {6^} for C", where e^ is the transpose of @,..., 0,1,0,. .., 0)
with " in the j-th place. For i < j, define linear maps 9ij G End{C'')hy
Oij(ej) = ei,9ij{ek) = 0,k ^ j; the matrix of S^j has a 1 in the ij-th place and
zeros elsewhere. The dij are eigenvectors of the action of Twith eigenvalues
expB7r\/^(Xi —Xj)), so x^ — Xj are eigenvalues for the action of L(T) on L{U(n)).

8 Chapter 1
(We are taking L{T) to be the diagonal matrices diag{\/—lyi,..., y/—^yn),yj €
R, and Xi € L{T)' is given by Xi{diag{^/^yi,..., V-^Vn)) = sZ-^Vi-)
u
SU{n)
S0Bn)
S0{2n + 1)
Sp{n)
: ±{xi- Xj)
: ± Xj ± Xj
: ± Xj ± Xj
±Xi
: ± Xj ± Xj
±2xi
l<i <j <n,
1 <i < j <n,
l<i <j <n,
l<i<n.
I <i <j <n,
l<i<n.
0
0
0
0
(ra — 1) times.
n times.
n times.
n times.
D
The Weyl group W acts on L{T) and permutes roots. If we regard the roots
as elements of L(T)', they form a configuration with great symmetry and very
distinctive properties, [1, Chapter 5, pp. 101-141]. The Dynkin diagram encodes
this configuration, as we proceed to describe. We may choose on L{T)' a positive
definite inner product invariant under W, so that we can talk about lengths of
roots, arag/e between roots and so on. For each pair of roots ±6, ker^ = ker(—^)
is a hyperplane in L(T) called a root plane. Conversely, it can be shown that
each root plane comes from only one pair of roots, ±9, [1, pp. 105-108]. The root
planes form a figure in L[T) called the (infinitesimal) Stiefel diagram.
X2 the root Xj—X2 is simple
the root X2 is simple
Fig. 2 The Stiefel diagram of 50E)
The root planes divide L(T) into convex open sets called Weyl chambers and
the Weyl group permutes the Weyl chambers in a way which is simply transitive,
[l,pp. 110-113]. We choose one and call it the fundamental Weyl chamber (FMIC),
C, shaded in Fig. 2. A root 9 is positive (respectively negative) i{ 9 > 0 on C
{9 < 0). A positive root is simple if it defines a wall of C. The Dynkin diagram

DeBnitions, examples and matrix groups 9
has one node for each simple root (i.e. for each wall of the fundamental Weyl
chamber). These nodes are joined by the following number of bonds:
0 if the roots are orthogonal
1 ... at 120°
2 ... at 135°
3 ... at 150°,
and these are the only possibilities, [I, pp. 118-121].
Examples 1.10. T, a torus, has the empty Dynkin diagram.
U{n). Take C to have Xi < X2 < ¦ ¦ ¦ < Xn-
-Xi+a;2 -a;2+X3 -X3+X4 -Xn-i+x„
We take the usual inner product of R".
SU(n) has the same Dynkin diagram as U(n), which is traditionally labeled
A„-i.
S0Bn). Take C to have —X2 < Xj < X2 < X3 < • ¦ ¦ < x^. (Dn)
Then the diagram is
X1+X2
^—X2+X3 -X3+X4 -Xn-l+Xn
-Xi +X2
S0{2n + 1). Take C to have 0 < xi < X2 < X3 < ¦ • • < Xn-i < Xn. En)
Xi -X1+X2 -X2+X3 -Xn-l+Xn
short
Sp{n). Take C to have 0 < Xi < X2 < X3 < • • • < Xn. (Cn)
2Xi -X1+X2 -X2+X3 -X„-i+Xn
long
Note: "short" (respectively "long") means Xj-xi = 1 < 2 Bxr2xi = 4 > 2).

10
Chapter I
We now have a rule which associates to a compact Lie group a Dynkin diagram.
However, as yet we only hope to distinguish between Lie groups with different Lie
algebras over R. To see this, note that we c?in recognise L(T) in L(G). But
it turns out that ?iny abelian subalgebra (that is, having zero brackets) of the
correct dimension i is an L{T). This action of L(T) on L[G) determines the
roots. In particulM, PSU{n) and SU{n) have the same diagram; they have
isomorphic Lie algebras and are locally isomorphic. Also, the Djmkin diagram
does not distinguish any abelian factors of G, since, as we have we observed before,
a torus has the empty diagram. The process of going from the Lie group to Dynkin
diagram is summarised in the following diagram.
Reductive
Lie algebras over R
Compact connected
Lie groups
®rC
Reductive
Lie algebras over C
Dynkin diagrams
In the diagram, the bottom arrow is factored round the square. (A lie algebra is
reductive if its adjoint representation is completely reducible.) The classical
classification scheme of Killing and E. Cartan is a classification of simple Lie algebras
over C and gives four infinite families [An, Bn, Cn, Dn) and five exceptions (G2,
F4, Ee, Ej, Ei).
For compact G, the real Lie algebra L(G) is the direct sum of its centre and a
semisimple ideal and this ideal is the compact form of its complexification: that is,
the Killing form of the ideal is negative definite (see Chapter 6 and [Ha, pp. 220,
193 and 215]). If G,G'are compact Lie groups, then Z;(G) is isomorphic to
L(G') if and only if L{G) ®r C is isomorphic to L{G') ®r C. Thus the Dynkin
diagram determines L(G) and hence, G, locally. In particular, corresponding to
each Dynkin diagram, there is a unique compact, connected, simply connected
Lie group. To each of the diagrams in the Killing-Cartan classification, there is
a unique simple Lie algebra, so a unique connected, simply connected, compact,
simple Lie group. The groups SU{n + l),n > 1, Sp{n),n > 3 correspond to
An and Cn, respectively; the groups Spin{2n + l),ra > 2 and Spin{2n),n > 4,
where Spin(r) is the simply connected cover of SO{r) constructed in Chapter 3,
correspond to Bn and D„. All these groups have rank n and are pairwise non-
isomorphic.
The purpose of these notes is to study the Lie algebras and simply connected
Lie groups corresponding to the non-classical (or "exceptional") Dynkin diagrams
(G2, F4, Ee, Et, Eg); these diagrams are listed below (see Chapter 9 for notation
and details).

Definitions, examples and matrix groups
11
?7.

Chapter 2
Clifford algebras
The references for this chapter are [2] and [11, Chapter 3, pp. 240-276].
In studying Spin{3) and SpinD) we use H = {i,j \ i^ = j^ = —1,1] = — ji),
and we could generalise H by defining a Clifford algebra to be the algebra with
generators ej,..., en and relations e^ = —1, 6^6, = — e,er, r ^ s, but it is better
to use a conceptual construction. For this, we begin with V, a non-zero finite
dimensional vector space over A = R or C.
Definition. T(V) = ® ®^V where ®°V = A.
n>0
We will write ®/ as V®". Define the product in T{V) as follows: if
Vi (S ¦ ¦ ¦ I® Vp e V^^ and lOi ® ¦ • ¦ ® u;, e V®' then their product is ?;) ® • •. ®
I'p ® lUi ® ¦ • • ® lu, e y®(P+'). For example, if V has a basis {x, y}, then T{V)
has a basis {1, x, y, xy, yx, y^, x^, ... }. (Here xy = x®y.) In general, T[V)
is a free associative algebra, called the tensor algebra of V and characterised by
the properties of Proposition 2.1.
Proposition 2.1. There is an inclusion V ^-^ T{V) and for any A-linear
map j : V -^ A of V into an associative A-algebra A, there is a unique homo-
morphism 0 : T{V) —> A extending j.
D
Suppose now we give V a symmetric bilinear form ( , ). (For example, on
V = A", set (u,v) = -u^v.) Let J = (v ®v - {v,v) ¦ 1 | v G V), a two sided
ideal in T{V) and put
Cl{V) = T(V)/J,
13

14 Chapter 2
the Clifford algebra over V with bUinear form ( , ). Write i : V —^ Cl{V) for
the composition V ^-> T{V) —>• T{V)/J = Cl{V). In practice we often omit the
notation for i and write v for an element of V or its image in Cl{V).
Proposition 2.2. (Universal property of Cl{V).) For any associative A-
algebra A and map j : V -^ A such that (jw)^ = {v,v) ¦ I & A, there exists a
unique map 9 : Cl{V) —> A of A-algebras such that 9i = j.
V ^—> Cl{V)
j\ / e
A
Proof. Immediate. []
Cl{V) is a Z2-graded algebra and T(V) a Z"*"-graded algebra. We put
T{V)o = e V®", T(V), = e V®^.
n even n odd
Then v®v - {v,v)-l &T[V)o so that J = Jo 0 Ji, where Ji = Jf\T{V)i and
Cl(y) = Cl{V)o © Cl{V)u where CZ(l/L = T(V)i/Ji.
If yl, 5 are graded algebras, we put (yl®B)n= © >lp®5, and (o®6)(o'®
6') = (-l)'^oo'®66' where a e /J^, 6 e B„ o' 6 A, b' e B,. Then A®B
becomes a graded algebra and we have associativity: [A®B)®C = A®(B®C).
Proposition 2.3. If V = V'®V" with {v',v") = 0 foraU v' e V, v" e V",
then
Cl{V)^Cl{V')(S)Cl{V").
Proof. We check that the right hand side has the universal property which
characterises the left hand side. Suppose given j : V —* A such that (jv)'' =
{v,v)-l, then V ^ V^ A induces 9' : Cl{V') -> A and V" ^ V^ A induces
6" : Cl{V") —> A. We now have to determine the relation between 6' and 6".
Step 1. If 7/ e V, v" e V", then {e'v')(e"v") = - (e"v"){e'v') in A.
Proof. {tJ + v",v' + v") = {v',v') + {v",v"), so j{v' + v") ¦ j{v' + v") =
JW) ¦ j{v') + j{v") ¦ j{v") and on expanding we have j{v')j{v") +j{v")j{v') = 0.
?
Step 2. If ?/ e V and x/l,...,v'^ e V", then
[e'v'w'inv':)) = {-ir{e"{iiv'nwv')-

Clifford algebras 15
Proof. We can expand 0"(Yli//) as 6" is a homomorphism. []
Step 3. K v[,...,v!peV', v'l, ...,v'^eV", then
(e'{m)){o"{nv'^)) = {-ir{e"{nv'^)){o'{m)).
Now define 9 : Cl(V') ® Cl{V") -^ A by x® y h^ (e'x){e"y). This is a map
of algebras by the definition of the product. The required isomorphism comes by
taking A = Cl{V). The inclusion V ^ V induces & : Cl{V') -» Cliy) and
V" ^ V induces 6" : Cl{V") -> CKV). Now take 6 : Cl{V')<SiCl{V") -^ Cl{V)
to be e{x ® y) = e'{x) ® e"{y). Q
Remark. Suppose V is 1-dimensional with basis {e}. Then Cl{V) has
basis {l,e}, because T[V) has basis {l,e',...} and J has basis {e^ —(e,e)l,
e' - (e,e)e, ...}.
Corollary 2.4. If dimA V = n and {ei,...,en} is an orthogonai basis for
V with {ei,ej) = \j5ij, then dimAC/(V) = 2" and {Of^y} is a basis wiiere ij
is 0 or 1. Q
Corollary 2.5. The maps A '-> C/(V) and V --* Cl{V) are mono.
Proof. Among the basis elements are 1, g],... , e^. Q
Corollary 2.6. Again assume that {e,} is a basi5 for V and (ej, ey) =
— Ey. Then the products in Cl{V) are determined by the foUowing relations:
el = -1, e^e, = - e^er, r jt s.
Proof. From the definition, we have e^ = — 1 and erfi^ = — erfij from
step 1 above. From these relations, we have
(n4*)(n4') = (-i)''ne^,
where 4 = it + jt (mod 2) and i/ = E jris- Q
Structure maps on Clifford algebras.
First there is the automorphism a : Cl{V) —> Cl{V) induced by —l:V—>V
(minus the identity). Thus a\Cl{V)o = +1, a\Cl{V)i = -1.
Example. C/(R) has generators 1, z, so a(l) = 1, a{i) = —i.
Cl{C) has generators 1, z, j, ij so a(l) = 1, a{i) = — i, a[j) = — j and
Q(ij) = ij
XV
Lemma 2.7. If { , ) on V is non-singular and x e Cl{V) is such that
= v(ax) for all v & V, then x is a scalar.

16 Chapter 2
Proof. Over A = R or C, we can diagonalise ( , ) and choose a basis
ei,...,en in V such that (e^es) = SrsK, K ^ 0. Now we can write x =
Jl^iTl^i, ^! ? A. Note that Ba is invertible as e,ej = Aj 7^ 0, so that if
/ i
xfij = 63(ax), we have ej^xe, = ax. But
[ (-1) ^^n^;^ i» = i
whilst aQJe]') = ( — 1) ' Hbj in all cases. Thus xe, = e3{ax) if and only if
J
A/ = 0 whenever i, = 1. Thus xe, = e3{ax) for all 5 if and only if A/ = 0
whenever ij = 1 for any 5. That is, A/ 7^ 0 only for / = @,..., 0) and x is a
scalar. []
Remark. If ( , ) = 0, then Cl{V) S A{V) = exterior algebra of V and
clearly x • i» = v{ax) for all i; ? V and all x in this case.
Definition. Define /3 : r(l/) ->• T(V) by /3(ui ® • • • ® Vn) = i-n ® • • • ® Wi.
This is an anti-automorphism {P{xy) = I3{y)l3{x)) and induces E : Cl{V) —>•
Ci(V) with I3\V = 1. AJso 7 = Q/3 = /3q : Cl{V) -> Cl{V) is an anti-
automorphism such that ylV = —1.
Example 1. Cl{R) is generated by 1, i, and /3A) = 1, P{i) = i, 7A) = 1,
7B) = -i.
Example 2. The algebra H = Cl{C) has basis 1, i, j, ij = k, and the
action of Q, /3, 7 is given by
a
P
7
1
1
1
1
i
— i
i
— i
J
-J
J
-j
k
k
-k
-fc
Note that for C/(R) = C or C/(C) = H, 7 is the usual conjugation.

Chapter 3
The Spin groups
The references for this chapter are also [2] and [11]. In addition see [BrJ.
Let V = A", A = R or C and ( , ) be the inner product {u,v) = u^t> and
form Cl{V) with respect to — ( , ). Let ei,... ,e„ be the standard basis in V,
so that in Cl{V) we have the products 6^63 [r < s ; there are n{n — l)/2 of
them) and these span a Lie algebra with [fire,, ?(?„] = erBaete^ — eieugre^. Thus
if r, s, t, u are different, [erej,e,eu] = 0 since e^ete^ = —ete,e^ = ete^Bs and
e,e3ete„ = ereteue, = eteuSr^s- If ^, h "^ are different, [e,.eu,e,eu] = 26^61 since
erfiueieu = —e^eieueu = erfii and e(e„ereu = ?(?, = -6^61. Finally, [6^6^,6^61] =
0. We wish to see this Lie algebra as the Lie algebra of a Lie group.
Definition. Pin(V) C Cl(V) is the subset of elements x such that
(i) xGx) = Gx)x = 1,
(ii) the map ttx : V —>• Cl{V) defined by Grx)i; = xv{j3x) maps V C Cl{V)
into V.
Example. Pin(R) = {z€Cl(R) \zz = l, ^^ eR} = {±l,±i}.
Proposition 3.1. (i) Pin{V) is a subgroup of the invertible elements of
Cl{V) and its Lie algebra is that given above (i.e. spanned by { 6,63 \r < s }).
(ii) The map 7r : Pin{V) —> 0{V), where 0{V) is the group of A-Iinear
maps V -^ V preserving —( ,), is an epimorphism with kerTr = {± 1}.
The closed subsets 7r-'(det~'(l)), 7r-'(det"'(-l)) of Pin{V) are in Cl{V)o,
Cl{V)i respectively and are connected for n > 2.
Explanations. We treat the real and complex cases simultaneously. If A =
R, 0{V) = 0{n) and if A = C, 0{V) is the complex orthogonal group, the
subgroup of GL{n, C) such that J2 z'^ is preserved. Linear maps J -.V -^V in
either of these groups must have determinant det/ = i:l, so 0A^) = det""'(l) U
det-\-l). If 7r-'(det-'(l)),7r-'(det-'(-l)) are connected, det-'(l), det-'(-l)
must also be connected.
17

18 Chapter 3
(Let us recall how to show that det~'(l) (= SO{V)) and det~'(—1) are
connected. It suffices to prove that SO(V) is connected by induction on n = dim V.
Clearly SO{V) is connected for ra = 1, so assume that 50A^) is
connected for dimK = ra — 1, w > 1. By appljnng [S, p. 377, Corollary 13] to
SO{n) — SO{n)jSO(n - 1), we see it is sufficient to show that S0(n)/S0(n-1)
is connected and this may be identified with {v & V \ (v,v) = 1}, n > 2, under
the map [A] h^ Ae„ where ej = @, 0,.. . , 0,1). For A = R, SO{n)/SO{n - 1)
is identified with 5""^ C R" and this is connected (w > 2) and for A = C, it
is identified with { ^ € C" | 7? H + zl = l]. Fnt z = x+ iy, x,y ^"EC and
consider the locus Re[z\-\ l-z^) = 1, that is x\-\ l-a:^=y?H Vy'i + l.
This is homeomorphic to S"~' x R", hence connected. On Re[z) = 1, we can
make ,/z into a well defined continuous function and by mapping (zj,... , 2^) to
(l/Jzf + ¦ ¦ ¦ + z^j[zi Zn), we get a retraction from the locus Re{z'f H h
z'^) = 1 to the locus zl + ¦ ¦ • + z^ = 1, so the latter is connected as well.) []
FinaUy, Spin{V) is the subgroup 7r-'(det~'(l)) = Pin(V)r\Cl{V)o. It comes
with a homomorphism n : Spin(V) —> SO{V).
Proof (of Proposition 3.1). It is straightforward to check that Pin{V) is
a closed subgroup of invertible elements of Cl{V). Next we show that 7r(x) e
0{V):
(Grx)i), Grx)u) = (Grx)v)^ = —xv{l3x){ax)v{^x) (as -Grx)i) = —(qx)i;Gx))
= —xvvy[x) = (v,v)x^x = (v,v).
This checks that 7r(a;) G 0{V). It is easy to see that tt is a homomorphism.
Suppose now that x ? kerTT. Then v = xvfix, for all x e K, so vax —
xv{Px)[ax) = XV. By Lemma 2.7, x is a scalar. But X7X = 1. Thus x^ = 1,
X = ± 1 and kerTT C {± 1}. Certainly ± 1 € kerTr, so keryr = {± 1}. Next we
need to estabhsh the following
Claim. LGr) : L{Pin{V) f] Cl{V)o) —> L{0{V)) is epi.
Proof (of Claim). One checks that Pin{V) n Cl{V)o is again a closed
subgroup. For r < s, {ere,){ere,) — -?^6^656, = -1 so x = e^^^''' = cosi -|-
(sini)erej is defined for R or C. Now,

The Spin groups
19
7r(x) =
/ 1
cos 2i — sin 2t
1
sin2f
cos2t
ly
(cos i + (sin <)eres j fiu (cos t — (sin O^rej j
if u 7^ r, 5
fcos2i + (sin 2i)erejjeu if u = r, 5.
Thus 7r(x) maps K to K, so x e Pin{V). In fact x ? Pira(K) nC/(V')o. We see
that the A-multiples of ?^6, lie in L[Pin{V) n Cl{V)o) and map under Z,Gr)
/o \
-2
0
to the A-multiples of
V
= Gr(x)) |t=o, which form a A-basis
0/
for L{0{V)), the skew-symmetric matrices. This proves that Z/Gr) is epi. []
It follows from the Claim that LGr) : L{Pin{V)) — i@(V')) is also epi; it
must be mono as kerTr is finite. Thus
L(Pin{V) n Cl(V)o) = L{Pin{V)),
and so L(Tr) : L{Spin{V)) —> L{0(V)) is an isomorphism. Since LGr) maps the
A-span of { CrCa I r < s } onto L[0{V)) and Z/Gr) is an isomorphism, { ere^ | r <
s } must be a A-basis for L(Pin(V)).
Now, using exp and log, we find that n maps a small neighbourhood of
1 in Ftii(V') n Cl{V)o onto a small neighbourhood of 1 in 0(V). Thus 7r
maps the identity component of Pin{V) n Cl{V)o onto the identity component
of 0A^), i.e. onto SO{V). But {±1} is contained in the identity component of
Pin(V) n Cl(V)o if n >2, since cosi + (sini)eie2, t e [0,7r], is a path from 1
to -1 in PinlV) n Cl{V)o. So the identity component of Pin{V)o r\ Cl{V)o =
¦K~\SO{V)) in Pin(V). Thus 7r''(det''(l)) is connected and contained in

20
Chapter 3
Cl{V)o- To show that 7r~'(det~'(—1)) is connected ?ind complete the proof, it
is sufficient to give an element in 7r~'(det~'(—1)) so that multiplying by it will
send 7r~'(det~'(l)) to 7r~'(det~'(—1)). The element ei will do, for one checks
that it lies in Pin[V) and covers the reflection diag{—1,1,... ,1). Q
Remark 1. Over R, the maximal torus T in Spin{m), m = 2ra or 2n + l, is
usually taken to consist of the elements H (cos 5Xr + (sin 5Xr)e2r-ie2r) in Cl{V),
r=l ^
Xr e R, which corresponds to
' COS Xi
sinxi
— sinxi
COSXi
C0SX2
sinX2
— sin X2
C0SX2
0
COSXn
sinXn
- sm Xn
COSXn
V 1/
where row ra + 1 has 1 as the last entry if m = 2ra + 1, and is empty for m = 2ra.
Remark 2. For A = R, we can show directly that every element in Spin{m)
is conjugate to an element in T from the corresponding result for SO{m), as
follows. Take g e Spin(m), ng e SO{m). Then there exists h e SO{m) such
that h{jtg)h-'^ e T C SO{m). Lift h to h' & Spin{m) under 7r : Spin{Tn) ->
SO(m) to get h'gh'-'^ e f.
(Alternative proof. For ng e SO{m), take a basis 61,...,6^ of R"* with
respect to which ng has matrix as in Remark 1 above. Then
{sm^Xr)b2r-ib2r) € Spin{m) covers g.)
n (cos \xr +

Chapter 4
Clifford modules and
representations
The references for this chapter are [8], [11] and [2].
Since 5^171A^) C Cl(V)o, any 0/A^H-010A1116 is a representation of Spin[V),
?ind some important representations of Spin{n) arise in this way.
Proposition 4.1. The algebras Cl{V) and Cl{V)o are semi-simple, so all
their representations are completely reducible.
Proof. Let { ei,..., e^ } be the steindard basis in V = A"* and consider
E = {± Yi s,' \ ij — 0 OT I}, a subgroup of order 2"^ of Cl{V), corresponding
to the matrices diag{±l, ...,±1) of 0{m). In Cl{V)o consider the subgroup
Eo of 2"* elements with J^h even, Eo C E C Pin{V). To avoid confusion,
i
write u for —1 e Cl(V)o when considered an element of Eo, E or Pin{y) \
u is the generator of kerTT. A module over Cl(y) gives a representation of E
in which u acts as —1. Conversely, a representation of E in which i^ acts as
— 1 gives a module over A{E)/{u + 1) = Cl{V), where A(E) is the group ring
over A oi E and A is R or C. The representation theory of Cl{V) can thus be
inferred from that of the finite group E and in particular all representations are
completely reducible. We argue similarly for 0/A^H, replacing E hy Eq. Q
Proposition 4.2. // dimV is odd, m = 2n + 1 say, then Cl{V)o has
one irreducible representation A of degree 2" affording a representation A of
Spin{2n + 1) with weights j (±Xi ± X2 ± • • ¦ ± x^), so there are 2" of these
weights. If dim V = m = 2ra, then Cl{V)o has two irreducible representations
A"*", A" of degree 2""' affording-representations A"*", A~ of Spin{2n) having
21

22 Chapter 4
weights
^(ixi ±X2 ± • • • ±Xn), (even number of - signs) for A'*', and
|(±Xi ± X2 ± • ¦ ¦ ± Xn), (odd number of — signs) for A".
There are 2""' weights. If A = C, these are complex-analytic representations of
Spinc{m).
Proof. By Schur's Lemma, u acts on any irreducible representation as either
1 or as —1. The ones in which it acts as 1 are representations of Eo/{i/) which
is an abelian group of order 2"*"^ so there are exactly 2"*"' 1-dimensional
representations of Eo in which u acts as 1. Since the kernel of Eq —> E^l {u) has
exactly two elements, the conjugacy classes in Eo are either one element (if the
element is central) or two elements ± g.
2n
Lemma. The centre of Eo is {± 1} if m = 2n+ I and {±1, ±Y[ei} if
1
m = 2n.
Proof. If we conjugate 5 = 0^/ ''^'^h e^es where v = 1. ^s = 0, we change
1
its sign; so if g is in the centre, j = ±1 or ±6162. . . 6^. The latter is in the
centre only for m even. []
Recalling that the isomorphism classes of irreducible representations (over C)
of a finite group are in 1-1 correspondence with the conjugacy classes, we see that
Eq has one (respectively two) more irreducible class(es) of representation(s) than
Eo/(l') if m = 2ra-I- 1 (respectively, m = 2ra). Let F C Eo be the subgroup
generated by 6162, ... , e2r-ie2r, • • • , e^n-iein ¦ This is an abelian group of order
2"+', so l^o ; F| = 2" if m = 2ra -f 1 and = 2"-' if m = 2ra.
Choose a complex 1-dimensional representation W of F in which u acts as
— 1 and e2r-ie2r sets as ie^, fir = il, i = %/""! ¦ Form
C(?'o) ®C(F) W' (i.e. the induced representation, IndpW),
where C(?'o) is the complex group algebra of Eg. This is now a representation
of Eo with degree 2" for m = 2ra -f 1 and 2""' for m = 2ra with u acting as
— 1. It has a basis consisting of
2n+l 2n+l
n ej' with Yl ji even, if m = 2n+l,
i odd=l 1
2n-l 2n-l
n ef with Y. ji even, if m = 2n.
i odd=l 1
When m = 2n -f 1, there are 2" choices for W (because we can choose n
signs ?r), Eo/F permutes them transitively and, by conjugating with e2re2r+i.

ClifFord modules and representations 23
we can change the sign of ?r without changing anything else. Each appears once
in C(?'o) ®C(f) W. We thus get a representation, A of Eo with character 2"
at 1, —2" at —1, and 0 elsewhere. By the orthogonality relations, A is an
irreducible representation of Eo. When m = 2ra, there are 2" choices for W
and under Eq/F they fall into two orbits: those with H^r = 1 and those with
n^r = —1- We can only change the sign of an even number of Sr: conjugating
with 62^628, r < s, changes the sign of both Sr and ?j. Hence we have one
irreducible representation of Eo which as a representation of F contains all the
W with e = n^r = 1 and another containing all the W with e = JJe^ = —1.
2n
The character of these representations is 2" at 1, —2" at —1, i^e at Yl^i,
1
2n
—i"? at —Ylci, ?ind zero elsewhere. This is an irreducible character by the
1
orthogonality relations. Thus these are inequivalent representations. Q
Remarks 1. These representations are self dual except in the case m =
2 mod 4 when the two representations are dual to each other, see Proposition 4.3
below. Here dual means complex conjugate.
2. If we take A = C and think of the complex form of Spin(V), we get
complex analytic representations. But to give the weights it is sufficient to take
A = R and consider the compact form of the group. This is what is usually meant
when we write Spin{V).
3. We calculate the weights of A for A = R as follows: The element
n (cosB7r)(|zr) + (sinB7r)(ix,))e2r-ie2r)
of the maximal torus acts on A with eigenvalues
n (cosB7r)(ix,) + (smBn){^Xr)){?ri)) = expB7rj f) k.x.)
n
and weights | JZ ErXr, where e^ = i 1- H
Proposition 4.3. The representation A ofSpin{2n+l) is self dual. The
representations A"*", A"" of Spin{2n) are self dual if n is even and dual to each
other if n is odd.
Proof. We have to prove the isomorphism in various senses of M' with N
where M, N are from A, A" as stated. Well, Eq acts on M' = HoTnc{M,C)
as [gh){m) = h{g-^rn), m e M, and Ci(C'")o = C(?'o)/(i'+ 1) acts ort M'
as {ah){m) = h({^a)m) where 7 : CT(C'")o —» CZ(C'")o is the
anti-automorphism of Chapter 2. From the proof of Proposition 4.2, we have an
isomorphism of the representations N and M* of the finite group Eq, which is an
isomorphism of C/(C'")o-modules. But Spinim) C C/(C'")o, so the isomorphism

24 Chapter 4
preserves the action of elements of Spin(m) provided this action is defined by
{gh)m = h[g~^Tn), which is the usual action. Note that this argument gives an
isomorphism of the representations of the complex Spin groups if we want. []
For the next result note that the inclusion A"* ^-> A"*"*"' induces an inclusion
C/(A'") ^-> C/(A'""'"') so we get Spin{m) "-^ Spin{m-\-l) covering the usual map
SO{m) ^ SO{m + 1), where A k+ ("J °).
Proposition 4.4. Under the inclusions
Spin{2n) ^ Spin{2n + 1) ^ Spin{2n + 2),
we have
^ A-
We will prove this proposition with the next one. We have Cl{A'')®\Cl{A'') =
Cl(h?''^'') which gives Spin(p) X Spin{q) -> Spin{j) + q) and Eo(p) x Eo{q) ->
Eoip + q). These cover the map SO{p) x SO{q) ^ SO(p + q) where {A, B) k+
f Q gj. Here the last map is an injection but the previous two are not: they carry
[u, v) to 1.
Proposition 4.5.
(i) Under Spin{2r) x Spin{2s) —> Spin{2r + 2s)
A+ ® A+ + A" ® A" <—I A+
A+®A- + A-®A+ <—I A".
(ii) Under Spin{2r) x Spin{2s + 1) —> Spin{2r + 2s + 1)
A+®A + A"®A <—I A.
(iii) Under Spin{2r + 1) x Spin{2s + 1) —> Spin{2r + 2s + 2)
A ® A <—I A+
A®A <—I A".
Proof. The proofs of the last two properties are closely similar. We have two
modules M, N to be proved isomorphic in various senses, namely the module
on the left and the pull back of the module on the right. In each case the weights
agree, so the characters agree on the compact group SpinR(p) x SpinR_{q).
For example, in the first case, if we take a weight j (ejXi + • ¦ • +er-i.sXr+s), with
ei?2 ¦ ¦¦?r-t-s = 1, then we get on the right Ksi^iH l-?ra:r) + |(?r+ia:r+i4 1-
Sr-i-sXr-t-s) either with ?iE2 ¦ ¦ .Er = 1 and Sr-i-l ¦ ¦ -Sr+a = ^ or with ?i?2 • • • ?r =

Clifford modules and representations 25
— 1 and ?r+i • ¦ • Er+a — ~^- This shows that the characters agree on the group
Spin^ij)) X SpinR(q). Hence the characters agree on the finite groups Eo(p) X
Eo(q) so the modules are isomorphic as representations of these finite groups and
thus isomorphic as modules over A{Eo(p) x Eo{q)). But they are isomorphic as
modules over A{Eo(p))/{u + 1) ®a A{Eo{q))/{i^ + 1) = Cl{A'')o ®a CKA')o • So,
the modules are isomorphic as representations of Spin/^(p) x Spinf^{q), where
5piraA(p) = 5pira(AP) with A = R or C. g
This gives several representations of SO(n + 1) since u € Spin{2n+ 1) acts
as (-1)®(-1) = 1 on A® A. We know that SO{m) acts on R"* = A' (where
A' is the i-th exterior power of dim G) ) and 50Bn) acts on A" = A'>+ + A"".
See [1, p. 177] and [12, p. 164]. The next theorem gives A"*" ® A~ in terms of
these exterior powers.
Theorem 4.6. (i) As representations of Spin{2n) we have
o-2(A+) = (A'>)++ A"-" + A"-* + • ¦ ¦,
A2(A+) = A"-^ + A"-* + ¦ • •,
a^A-) = (A")-+ A"-" + A"-* + • ¦ •,
A2(A-) = A"-^ + A"-* + ¦ • •,
A+ ® A" = A"-' + A"-' + A"-^ + ¦ ¦ ¦,
where in each summation we have non-negative exterior powers,
(ii) /Is representations of" 5pmBra + 1) we have
a\A) = A" + (A"-' + A"-") + • • ¦ + (A"-"*-^ + A"""*-") + • ¦ ¦,
A2(A) = (A"-' + A"-^) + (A"-^ + A"-«) + • ¦ ¦ + (A"-"'-' + A"-''*-^) + • • • .
where again in each summation we have non-negative exterior powers.
Note. For a representation V, cr^(V) is the second symmetric power, see
[Laj.
Proof. We will prove just the result for A'*' ® A". First we establish the
sum of the last two results, that is for A ® A, which has Spin[2n + 1) acting on
it as g{m®Tnl)=gm®gm'.
Since A is self dual, we have A®ASA®A' = HomciA, A); g acts on
h e /fomc(A, A) to give ghg"^. Since Cl{V)o is semi-simple and A is its only
irreducible representation up to isomorphism, we have /fomc(A, A) = Cl{V)o
with g acting on a € Cl(V)o by gag~^. This is the same as the action of G we
know already: G = Spin{2n + 1) acts on V and that induces automorphisms
of Cl{V)o ¦ (The proof of this is that since g e Pin{V) n Cl{V)o, we have

26 Chapter 4
g~^ = ^g = Pg.) But this representation of G on Cl{V) splits as can be seen as
follows. We define a map V®* -v Cl{V) by
V. ?(o-)
Vi (g) V2 B) ¦ ¦ ¦ <S> Vi >-* J2 -rr- v^n\ ® v^m ® ¦ ¦ • ® ^^iTCi)
<76Ei t!
(where E; is the symmetric group) which leads to vector space isomorphisms
®\'V^Cl{V) and ®\^'-{V)^Cl{V)o. Hence A(S)A = X°+\'^ + \* + --- + X'^''.
But A" = A-* when i + j = 2ra + 1, so A ® A = A" + A' H + A", which is the
sum of cr^(A) and A^(A).
The same argument works for SpinBn), except with two irreducible
representations A"*" and A~ we have:
/fOTnc(A+, A+) + HomciA-, A") S Cl{C^'')o = A" + A' + A" + • • • + A^".
First consider n odd so that A" and A"*" are dual to each other. Then
2A+ ® A" = A" + A^ + A" + ¦ • ¦ + A^" = 2A'>-' + 2A'>-^ + • • ¦ + 2.
When n is even, we have
A+ ® A+ + A" ® A" = A" + 2A'>-^ + 2A'>-' + • • • + 2A''.
On the other hand, we can take the formula for A ® A and restrict it from
Spin{2n + 1) to SpinBn) and get
(A+ + A") ® (A+ + A-) = A" + 2A"-' + 2A"-^ + • • ¦ + 2\°.
Subtracting off the formula for A"*" ® A'*' + A~ ® A", we have
2(A+® A-) =2A"-' + 2A'>-3 + --- + 2A',
so
A+®A- = A"-' + A'>-3 + --- . Q
Corollary 4.7. There is a bilinear map fi : Mi^ M —> C which is invariant,
non-zero and
symmetric in cases M = A'*', A~ if 2n = 0 (mod 8),
M = A if 2ra + 1 = 1 or 7 (mod 8);
anti-symmetric in cases M = A'*', A~ if 2ra = 4 (mod 8),
A/ = A if 2ra -fl = 3 or 5 (mod 8).
Proof 1. From Theorem 4.6, we can see where a siunmand A" = C arises.

Clifford modules and representations 27
Proof 2. We always have an evaluation map M' ® A/ —>• C. When M is
self dual, this gives an invariant non-zero map M ® M —> C. When M is C-
irreducible, this must be either symmetric or anti-symmetric but not both. Thus
we can find out which case holds by testing on one pair of vectors. Since fi ^0,
there are eigenvectors of the T-action such that fj,{w ® w') ^ 0. Moreover, u;,
w' must be eigenvectors of opposite weights, for if tw = \(t)w, tw' = y[t)w',
then fj,(w ® w') = fj,{tw ® tw') = fj,(\{t)w ® \'{t)w') = \{t)\'{t)/j,{w ® w') and
\{t)\'{t) = 1 for all t e T.
Any w' of weight opposite to w will do since all are scalar multiples of one
another. Let us construct one by taking
e = 616365 ... e2n+i Bra+l=3mod 4), and
6 = 616365 ... e2n-i in all other cases.
As 0 5^ fj,{w®ew) = fj,[ew®e''w), e^w must be an eigenvalue for the same weight
as w, so we must have e^w = \w for some scalar A. So, 0 ^ ii(w ® ew) =
Xfj,{ew ® w) and fi is symmetric if A = 1, anti-symmetric if A = —1.
If 6 = 616365... 62„_i, then 62 = (-lM'>("+J) =1
If 6= 616365... 62„+i, then 6^ = (-l)j('>+l)(n+2) =
1 if 2ra = 0 mod 8,
-1 if 2ra = 4 mod 8.
1 if 2n+l = 7 mod 8,
-1 if 2ra-fl = 3mod 8.
D
Corollary 4.8. The following representations of the compact groups
Spin^[m) are
(o) reai : A+, A" if m = 2ra = 0 (mod 8),
A if 7n = 2ra-hls 1,7 (modS).
F) symplectic : A'*', A~ if m = 2n = 4 (mod 8),
A if 7n = 2ra-l-1 =3,5 (mod8).
First Proof. The conclusions of Theorem 4.6 are sufficient to show that
M = A"*", A", A comes from real/symplectic representations of the finite group
Eo, i.e. there is an action of R(?'o) on a module over R or H which affords the
given representation. But R(?'o)/(i'+ 1) — Cl{V)o acts on M and SpinR{V) C
Cl(V)o, so SpinR{V) acts on M.
Second Proof. The argument for finite groups works for compact groups
and the first sentence of the first proof suffices (with 'compact' replacing 'finite').
D

28 Chapter 4
The Theorem of Weyl on R{G}.
We end this chapter by describing how A, A""" and A" fit into general
representation theory. For example, when G = S0E), we can plot the weights of
the adjoint representation as follows:
—Xi + 12 •
—Xi •
-II - 12 •
• X2 • Ii + X2
• 0 • Xi
• — 12 •11—2:2
1
1 1 1
1
A^ = adj :
1 1 1
1 2 1
1 1 1
The roots are the non-zero weights here and the weights of any representation
are linear combinations of these. Thus if A' is the i-th exterior power of the
standard representation of SO E) on R^ then the weights of various representations
are as follows:
A'
where if Wj is a root (or zero) and j an integer we have replaced JW; by j in
the diagram. As a further example, we list the diagram for the second symmetric
power a^ of the standard representation of R^ :
1
1 1 1
a^ = ff2(R5) : 113 11
1 1 1
1
These patterns must be invariant under the Weyl group and, for irreducible
representations, aU non-zero entries lie within the convex hull of the orbit of
one with unit multiplicity. The irreducible representation is completely
determined by giving that orbit or even one weight in it. To make this quite
explicit we must choose a fundamental dual Weyl chamber (FDWC) where FDWC
= {he L{T)' I F1,/i) > 0 for each simple root 61}, [1, p. 125].
Let R{G) be the representation ring of G (the free abelian group generated
by the isomorphism classes of irreducible, complex representations of G). Then
we have
Theorem 4.9. (H. Weyl) If G is compact, connected and simply-connected
then R{G} is a polynomial ring with generators in 1-1 correspondence with the
nodes of the Dynkin diagram as follows. For each simple root di, define the
corresponding edge of the FDWC by {4>,di) > 0 and {<f>,9j) = 0 for all other
simple roots dj . Go along the edge to the first weight and take the corresponding
representation S;. This is the generator [1, p. 164]. Q

Clifford modules and representations 29
Example. G = Spin{2n). Let A' = A*(R^")®C. Then the Dynkin diagram
is
A+
>A' A^ A'
. . . a • a
labeled as shown, so
R(SpJnBn)) s Z[ A+, A", A', A^ ..., A"-^].
Here for example, the weight for A""" is ^{xi + ¦ ¦ ¦ + x„), corresponding to the
root Xi + i2- The formulae in Theorem 4.6 are those for writing A"~\ A""*", A"~
in terms of these generators.

Chapter 5
Applications of Spin
representations
Construction of Gj.
For large values of n, the diagrams of type A„, B„, C^ and Dn are distinct.
For small values of n we have the possibility of exceptional isomorphisms between
the classical groups as follows.
(a) SpJnF) = SU{A). Both have dimension 15, rank 3 and the Dynkin
diagram
(b) Spin{5} = Sp{2). Both have dimension 10, rank 2 and the Dynkin
diagram « ¦ .
(c) 5pmC) s SU{2) ^ Sp{l) = S' cH.
(d) Spin{4) ^S' xS\
We just prove the first two isomorphisms ; the others are easier.
Proposition 5.1. We have the following isomorphisms of Lie groups:
SpJnF) s SU(i), Spin{b) = SpB).
Proof. We do both cases in parallel. Spin{b) has the representation A
of degree 4 over C and degree 2 over H. We can impose a Hermitian form,
invariant under the compact group Spin{5), giving us a homomorphism A :
Spin{5) —^ Sp{2). Similarly Spin{6) has the representation A""" of degree 4 over
C and we have A"*" : Spin{6) —» C/D). We first wish to show that ImA'^ C
SU{i). Let i e T C Spin{6). Then t acts with eigenvalues defined by weights
{ ^(ii + 12 + 13), \{xi - X2 - 13), 5(-ii +X2- X3), K-ii -X2+ X3) } which
add to zero so the eigenvalues multiply to 1 and t must act with detl. Hence
31

32 Chapter 5
any gtg~^ acts with det 1 and A"*" maps to SU{i). Now A is faithful: if
g € SpinBn+ 1) and p i-* 1 in Homc{A,A) = Cl{V}o, then g is 1. Also
A+ is faithful if n is odd, for if g acts as 1 on A"*", then g € Spin{2n) acts as
1 on the dual, A", so p i~* 1 in Homc(A+, A+) + HoTnc(A~, A") = Cl{V)o.
Thus g = 1. Hence the two maps A, A''" are mono and induce monomorphisms
L(A), I/(A+) and so are iso for dimensional reasons. Thus A and A''" map small
neighbourhoods in Spin{5), Spin{6) onto small neighbourhoods of Sp{2), SU{4)
respectively. But Sp{2} and SU{i) are connected, hence A : Spin{b) —> Sp{2)
and A""" ; SpjnF) —+ Sf7D) must be epi. Q
Corollary 5.2. The group Spin{b) acts transitiveiy on the unit sphere
S^C A.
Proof. Sp{2} acts transitively on 5^ C IP. Q
Corolkiry 5.3. The group Spin{6) acts transitively on pairs {x,z), x €
S^ C R®, z € S^ C A+. Moreover, the subgroup fixing z € S^ C A+ may fae
taken, by a suitable choice of z, to be SU{3) C Spin[6), where this inclusion
arises by lifting the composite SU{3} ^-^ U{3} ^^ S0{6} to Spin{&).
Proof. SpinF) covers S0F} which acts transitively on S^. Taking a
suitable X, e.g. I = @,... , 0,1), the subgroup fixing x is 5pjnE). The restriction of
A'*" to SpJnE) is A and SpJnE) acts transitively on z € S^ <Z A by the last
corollary. The weights of A"*" are {5B:1 + X2 + X3}, ^{xi — x^ — 13), ... } and
their restrictions to T C SU{3) are {Q,X\,X2,X3 }, hence the restriction of A"*"
to SU{3) is 1 + A'. So this SU{3) fixes points of 5^ C A+: in fact, a whole
circle of them. For any such fixed point, the subgroup fixing it is no bigger, since
in SU{i) = SpjnF) the subgroup fbcing a unit vector in C is an SU{3). Q
Corollary 5.4. The group Spin{7) acts transitively on triples {x,y,z)
where x, y are orthogonal in S^ C R^ and z € S' C A.
Proof. SpinG) covers S0{7) which is transitive on points y € 5*. Choose
y = @,...,0,1). Then the subgroup fixing y is Spin{6). Now A restricts to
A"*" + A" over C, but starting with A as a vector space of dimension 8 over R,
the restriction to SpinF) is the representation of dimension 8 over R underlying
A"*" (or A~). Finally, note that by the previous corollary, Spin{6) is transitive
on pairs (i, z), i € S^ z € S^ C A+. Q
Theorem 5.5. Consider the subgroup G of SpJnG) which fixes a point
z ? S^ C A. Then G is a compact, connected, simply-connected. Lie group of
rank 2, of dimension 14, with the Dynkin diagram « » and commonly called
Gj. Moreover, G^j is transitive on pairs {x,y) of orthogonal vectors in S® C R^.
Proof. The last sentence follows from Corollary 5.4. Clearly G is a closed
subgroup of Spin{7} and since Spin{7) is transitive on S^, we have dimG =

Applications of Spin representations 33
dim Spin{7) — dimS^ = 21 — 7 = 14. Let H C G he the subgroup fbdng
y = @,... ,0,1). Then H is the same as the subgroup of Spin{6) which fixes
z and by a suitable choice of z, we can take H = SU{3} C SpinF). Since H
is connected and G/H = S* is connected, we find that G is coimected, (use
the homotopy exact sequence of the fibration H -* G -* G/H, [S, p. 377)).
Similarly G is simply-connected. Next we determine the roots of G. We know
how H = SC/C) acts on L{H) C L{G), namely by the adjoint action, so next
we wish to know how H acts on L{G)/L(H), the tangent space to 5^ at j/.
By construction, S® = Spin{7)/Spin{6), so we need to look at the action of
SpinF) on L{Spin{7})/L{Spi'n{6)). We know the weights of this action, namely
{ill, ^Xj, ±3:3 }. This is the geometrically obvious action where the tangent
space to S^ at @,... , 0,1) is R®, the space of the first 6 coordinates and Spin{6)
acts on it as usual. Thus T C SU{3) acts on L{G) with weights {0, 0, ± (ij —
X2), ±{x2 — X3), ±A3 — 11), ±ii, ±X2, ±13}. We conclude that T is maximal,
these are the roots of G, and the Dynkin diagreim is « » . []
Gj starts life with two obvious representations: a 7-dimension?j
representation G2 C SpJnG), acting on R'' with weights {0, ixi, ixj, ±13} and a
14-dimensional representation. Ad with weights as in Theorem 5.5. These are the
two generators in the H. Weyl theorem (Theorem 4.9) applied to ^^(Gj) .
SpJn(8) and triality.
We turn next to Spin{8) and some background. If G is a Lie group we can
form
(i) Aut{G}, which is another Lie group but not necessarily connected even if
G is.
(ii) Inn{G), the inner automorphisms given by Qg(/i) = ghg'^. This is a
normal subgroup of Aut{G).
(iii) Out{G) = Aut{G)/Inn{G).
If Q G Aut{G} and p : G —> Aut{V) is a representation, then we can form
G —> G —> Aut{V) and if a € Inn{G) then pa is equivalent to the
representation p. Thus Out{G) permutes the equivalence classes of representations of G.
In particular, Out{Spin{8)) = E3 which permutes the three basic representations
of Spin(8), see Theorem 5.6 below.
Example. The (n -|- l)-dimension?j representation A" of SU{n + 1) may
be interpreted as SU{n + 1) —> SU{n + 1) and this takes A' to A".
Example. Define q : S0Bn) -^ 50Bn) by a{B) = ABA-^ where A =
I ^"~ _, I 0 S0Bn). This Q is usually called "the outer automorphism of

34 Chapter 5
S0Bn) which becomes inner in 50Bn + 1)" ; in S0Bn+ 1), we can conjugate
by (fiap(l,..., 1, —1, —1). The resulting automorphism Q lifts to Spin{2n) as
conjugation by ejnesn+i in SpinBn+ 1) or by ejn in Pin{2n+ 1). If T is the
standard maximal torus in SpinBn), then aT = T" and 0B;;) = Xi for I < i <
n — 1, q(x„) = —Xn, so that Q interchanges A"*" and A". Therefore it cannot
be an irmer automorphism. More generally, if a : G -* G is an automorphism
of a compact, connected. Lie group and T a maximal torus, then aT is a torus
in G, so by suitable conjugation, aT = T and after conjugation with a suitable
element of N{T), we can assume that q preserves the fundamental Weyl chamber.
Thus a permutes the walls of FWC and induces an automorphism of the Dynkin
diagram, which is to say that it permutes the representations attached to the nodes
of the diagram.
Example. An = SU{n + 1).
Here A* ^ A"+'-'
D„, n > 4.
A"
A^ A'
Here the roots marked A* are interchanged, keeping the others fixed.
Di = Spin{8).
Here dimRA"*" = dim^A = 8 and G has exactly three classes of
representations of degree 8, namely A""", A", A'.
Theorem 5.6. Out{Sjnn{8}} = T.3 acting on A+, A'andA'.
Proof. An automorphism a : Spin{8) —»• Spin{8) gives an 8-dimensional
representations of Spin(8). Conversely, given any 8-dimensional representation of
Spin{8) on V, we can impose a positive definite inner product, pick an orthogonal
basis, get a homomorphism Spin{8} —> 0(8) and remark that since Sptn{8) is
connected and simply-connected it maps into 50(8) andhftsto Spin(8). But the
classification of automorphisms is by conjugation in Spin{8) or S0(8) wherezis
the ciaissification of representations is by conjugation in Pin{8} or 0(8). For
example, taking A"*" : Spin(8} —>¦ Spin(8}, we certainly have A+ *~i A' but either

Applications of Spin representations 35
(A' ^-l A"*", A" *-i A~ ) or (A~ ^-l A"*", A' *-i A" ) and both possibilities can arise.
If one arises, we get the other by composing with some a : Sptn(8) —+ Spin{8}.
Similarly for A" : Spjn(8) —^ Spin{S).
Thus the map Out{Spin{8)) —+ E3 is onto. If some 6 : Spin{8) —+ Spin{S)
maps to the identity in E3, then <5*(A') = A', so 6 is conjugate to 1 by an
element in Pin(8) of det ± 1 and since E*A"*" = A"*", it must be an element of
det 1 so E is inner and Out{Spin{8)} —> E3 is mono, hence an isomorphism. See
[6] and [5], ?
When talking of An = SU{n + l), we call the symmetry of the diagram duality,
A' ® A" S A"+' = C"+\ and when talking of Spin{8), we call the symmetry
triality.
Definition. Let Vi, V2, V3 fae reaJ vector spaces. Then a triality is a iinear
map / : Vi ® V2 ® V3 —» R such that for any non-zero f] G Vi, V2 € Vj there
exists 113 e 14 such that f{vi ® fj ® us) 7^ 0 (and similarly for 111,113 y^ 0,
i'2>i'3 7^ 0 j. If each Vi has an inner product, we say that f is a normed triality
if [/(^i ® i'2®i'3)| < llnill ll^sil ||i'3l| and for all 111,1127^0 there is a 1K7^0 for
which the bound is attained (and similarly for the other two cases).
Examples, (i) Vi = 14 = V3 = R, C or H respectively and take f{x ®
y®z) = Re(xyz).
(ii) Vi = A+, V2 = A", V3 = A' and / as in Proposition 5.7 below.
These examples are related as follows. Suppose given a triality /. Then for any
U] 7^ 0 we get a duality of V^, V^, so we must have dim Vj = dim V^ and similarly
dimVi = dimVj = dimVs. We can transpose V3 to get /' : VjigVa —+ V^. Choose
ei 7^ 0 in Vi and use it to identify Vj with V^ by Uj ^-^ /'(ej ®V2). Similarly,
choose 62 7^ 0 in Vj and identify Vi with V^. We now have /" : V^' ® V^' —* V^*
for which /"(ej ® ej) acts as a two-sided unit and /" is non-singular in that if
I, y 7^ 0 e V^', then f"{x ® y) 7^ 0. So we have a division algebra over R and
consequently dimK = 1,2,4 or 8. If we start with a normed trisjity, we get a
normed algebra.
In example (i), for R, the symmetry group of this triality is — ^ ^^ —
S° X S°; for C the symmetry group is ¦?' x ¦?! x 5' ^ j^2. f^^ jj ^.j^g group is
{±1} ¦
Proposition 5.7. There is a non-zero trilinear map f : A"*" ® A" ® A^ —> C
invariant under Spin{8) and unique up to a non-zero scalar multiple. If we
work with real representations, we may insist that f be real valued smd fox
I 6 5^ C A+, 7/ e 5^ C A-, z e 5^ C A' = R' we have
-l<f{x®y®z}<l

36 Chapter 5
where both bounds are attained. The map f is then unique up to a factor ± 1.
Proof. By Theorem 4.6 , we have A+igtA" = A' + A^ Here A' and A' are
irreducible, A' is dual to A' but not dual to A^. Therefore
A' (gi A+ (gi A- = A' (gi A' + A' ® A^ s Homc{\\\^) + Hcxmc{\\ A').
Only the first siunmand maps Spjn(8)-invariantly onto C, proving the first
part of the proposition. But S^ x S^ x S^ is compact, so putting
c= max |/(l®2/®z)|
!NI=!|y!l=!|z||=l
we have —c < f{x®y®z) < c. The bounds are attained because we can change
the signs of x, y and z. Also c > 0 since / 7^ 0. By multiplying / by a non-zero
constant, we can assume c = 1 and this is attained. Q

Chapter 6
The exceptional groups:
construction of Eg
We draw the following lesson from our construction of Gj: To describe an
unknown group G, it is useful to find a known subgroup of maxim?j rank H C G
and to give an account of G/H.
Theorem 6.1. There exist Lie groups G with subgroups H as specified in
the following table.
local t3rpe of H
Spin{9)
5pJnA0)xC/(l)/Z4
5pJnA2)xSp(l)/Z2
Spjn(l6)/Z2
rank
4
6
7
8
dim
36
46
69
120
L{G)/L{H) as C rep.
A
A+®^^ + A-®^-^
A+(giA'
A+
dim
16
32
64
128
G
Fa
Ee
Ej
E»
rank
4
6
7
8
dim
52
78
133
248
Notes. 1. In SpJnA6)/Z2 the Zj is generated by H^i which acts on A"""
1
as i' = 1 and generates ker A""".
12
2. In 5pJn(l2) x Sp(l)/Z2 the Zj is generated by (Hei, —1) which acts
1
on A+ ® A' as i^ ¦ (-1) = 1.
10
3. In SpJn(lO) X U{l)/Zi the Z4 is generated by {Ubj, i) which acts on
1
A+®^' + A'®^-^ as i^®i^.
The first column to be filled is that for dim(L(G)/L(/f)) and then we proceed
to find groups which have representations A or A""" of these dimensions. We
begin with the construction of the Lie algebra of E^, look for subgroups in the
next chapter and finish the proof of Theorem 6.1 in Chapter 8.
37

38 Chapter 6
Construction of a Lie edgebra of type E^.
For ?^8, there is no representation of smaller degree than Ad, so we use that.
Take L + A""" where L — L{Spin{16)) and consider this simultaneously over
R and C. For a while we can work with Spin{2n). Now, by Proposition 3.1, L
C C/Bn)o has a basis {6,63 \r < s}, and A"*" is a representation of Spin{2n)
or L, over R. That is, for all a e L, u G A*' we have [a,u] ? A+ satisfying
the Jacobi identity [[a,6],K] = [a, [6,w]] — [6, [a,w]] and in fact [a,u] is just
the usual multiplication of li € A+ = C{Eo) <8c(F) W (Proposition 4.2) by
a e CZBn)o. Assume now that 2n = 0 (mod 8) and consider A"*" as a real
representation of Spin{2n). Choose fi : A"*" ® A"*" —+ R, a symmetric, bilinear,
non-zero map invariant under Spin{2n), i.e. {gu,gv) = {u,v) for g & Spin{2n),
u,v G A''", see [1, p. 36]. The linearised form of this is: ([a,u],v) + {u, [a,v]) =0.
(This is the equation to be used whenever we say that a bilinear form is invariant
under a Lie algebra.) The elements of L can be considered as skew-symmetric
matrices (see the proof of Proposition 3.1) and on matrices we have an inner
product {A, B) = Tr{AB) which is a symmetric bilinear form, real on real
matrices. The invariance property for X € GLBn) is Tt{XAX'^XBX'^) =
Tr{AB). If we put X = I + tY and pass to the limit, we obtain the linearised
version, Tt{[Y,A]B + A[Y,B\) = 0, so we get {[Y,A],B) + {A,[Y,B]) = 0.
Under this identification, e^e, corresponds to the matrix with all entries zero
except in positions (r, s) and (s,r) where we have respectively —2 and 2. This
gives F^65,6^63) = —8 and to remove this undesirable factor, we put {A,B) =
— ^Tr{AB) so that now (eres.ejeu) = ErtEsu- In the next lemma, we transpose
the action L ® A"*" —» A"*" above to get a map A""" ® A''" —> L.
Lemma 6.2. For all u,v € A"*", there is a unique [u,v\ € L such that
(a, [u,v]);, = ([a,u],v)^+ for all a € L and [u,v] is bilinear in u, v.
Furthermore, if u, w e C igi A""" are such that
n
627-165,1'= JK for all q (corresponding to a weight 2S^t),
1
e2q-.ie2qU! =—iw {or all q (corresponding to Si weight — ^JZii),
and {v,w) = 1, then
(i) [v, w] = 1F163 + 6364 +¦¦¦+ 62„_i62„);
(ii) [62,62,1;, u;] = F2,_i-Hi62,)(e2r-i-H 162,.), q<r\
(iii) [62,162^2 ... 62,2„w, w] = 0 if m > 1 and 51 < 52 < • • • < 92m-
Proof. Clearly ([o,w],ii) is a linear function of a and since the inner
product on L is non-singular, we must have {[a,u\,v) = (a,b) for some 6 =
[v.,v\ ? L. Since ([a,u],ii) is bilinear in u, v, so is b. Everything so far works
over R or C. We proceed to derive the explicit formulae.
(i) First we have F2,-162,^, w) = (iu, lu) = i and (e^esV,w) = Q if 6,6, is

The exceptional groups: construction of Eg 39
not one of the basis elements 62,-162^. Thus [vjw] is paired to i if a = ?27-162,
and to 0 for all other basis elements.
(ii) For simplicity of notation consider [626411, w]. Then Fr6s6264ti, w) =
0 except when {r,s} = A, 3), A,4), B, 3), or B,4), when we get respectively
l,i,i,—1. This yields [626411, w] = 6163 + 16164 + 16263 — 6264. Notice that 626411
has weight ^{—Xi — 0:2 + a;3 H + x„) while w has weight |(—ii — • • • — x„)
so that [6264I1, w] must have weight —ii — 12- In fact, 6; + ie2 has weight —Xi
and 63+164 has weight —x^.
(iii) It suffices to note that for all 6,6^, r < s, F,6362,162,2 ¦ • • B2q2^v, w) = 0.
D
Remeirk. The map [ , ] : A""" ® A""" —» L is invariant under Spin{2n)
because everything in the construction is invariant under Spin{2n). The linearised
form of invariance (that is invariance under L) is
[a, [w.u]] = [[a,ii],i;] + [w, [a,w]]
and this equation is established as follows.
It is sufficient to show that for all b Q L
i-b,[a,[u,v]]) + (b:[[a,u]M) + {bAn,[a,v]])
is zero. This expression, using the invariance of ( , )i, under L and the definition
of {u,v], is
{[a,b],[u,v]) + {[b,[a,u]],y) + {[bM.['^M)-
Using the invariance of ( , )a+, this is
([[a,6],K],i;) + ([6,[a,wl],t;)-([a,[6,K]],i;) = 0,
using properties of the action of L on A"*". Q
We now proceed to give L+A"*" the inner product with L and A""" orthogonal
and
(a + ti, h + u) = @,6);, + (ii, v)a+, for all a,6 € L, w, v € A""".
The Lie bracket [a, 6] is as in L, [a,u] as the action of L on A'*", [u, a] = — [a,u],
and [u,v] as in Lemma 6.2.
Theorem 6.3. // 2n = 16, L +A""" becomes a Lie algebra with an invariant
inner product.
Proof. The inner product is invarisint under L by definition, and under A"*"
by the definition of [w, v]. We need to prove anti-commutativity and the Jacobi
identity. Clearly [a, 6] = —[6, a] since L is a Lie algebra, and we define [u, a] to

40 Chapter 6
be —[o,w]. To see that [u,v] ~ —[v,u] observe that for all a & L, u,v & A""",
we have
{a,[u,v] + [v,u]) = {[a,u],v) + {[a,v],u)
= ([a,u],v) + {u, [a, v]), by symmetry of ( , ),
= 0, by the invariance of ( , ) under a & L.
Since this is true for all a, we have [u, v] + [v, u] = 0.
For the Jacobi identity, we need to discuss several cases.
(i) Three variables in L, none in A"*": The identity will hold here since L is
a Lie algebra.
(ii) Two variables in L, one in A*: We have [[a,b],u] = [a,[b,w]] —
[6, [a, u]]=Q. Thus [a, [6, k] ] + [k, [a, 6] ] + [b, [u, aj ] = 0.
(iii) One variable in L, two in A''": Here, by the invariance of the bracket
under L, [a, [u,v]] = [[a,u],v] + [u, [a, v]], which leads to the Jacobi identity.
(jv) All three variables in A'*": This is where we have to use the fact that
n = 8. We set down a genersJ procedure for checking identities of this type by
using symmetry.
Standard operating procedure.
Step a. Express the identity to be verified as the vanishing of a G-map
f : U —* V where U, V are representations of the compact, connected. Lie
group G.
Example. G = SpinA6), / : A+(giA+1S1A+-+A+, x®y®z^[x,[y,z]\ +
Step b. It is sufficient to check that / = 0 on a basis for U consisting of
eigenvectors of the T-action where T <ZG is a maximal torus.
Step c. If f{u) = 0 for one eigenvector u & U, then the result follows for
each nu, n € N{T). (Proof: f(nu) = nf{u) — 0 and nu is an eigenvector
since t{nu) = n{n'^tn)u = nt'u = n\{t)u = X{t)nu.) We need only check on
eigenvectors corresponding to one weight in each orbit of the Weyl group.
Thus it is sufficient for us to check the Jacobi identity when x, y, z are
eigenvectors of T. Suppose
X has weight | J2 ^r^r,
y |E/9r2;r,
Z ^JllrXr,
Qr = ±l
/3r = ±l,
7r = ±l.
Thus f{xiS)y®z) is an eigenvector with weight 5 Y, {(^r + Pr +'yT)Xr and we
can assume f{x ®y®z)=Q unless 5 E(«r + Pr + lr)Xr is one of the weights.

The exceptioneil groups: construction of Eg 41
2 ll^T^r, ?r = ± 1, which occuT in A"*", that is, unless for all r, two of Qr, /?r, 7r
are s^ and one is — e,. We now have four cases to consider.
Case 1. One of x, y, z has all 8 occurrences of — ?,.
Case 2. There are 6 occurrences of — e, in one of x,y,z, 2 in another, 0 in
the third.
Case 3. There are 4 occurrences of — Sr in one of x,y,z, 4 in another, 0 in
the third.
Case 4. There are 4 occurrences of — e, in one of x,y,z, 2 in another, 2 in
the third.
These are all the possibilities since the Weyl group permutes ij,... ,a;8 in any
way and changes an even number of signs. In any case, it does not matter which
of the three vectors is x, y ox z because f{x ®y®z) is anti-symmetric in x, y,
z by the anti-symmetry of [ , ].
Case 1. We may assume that all eight coefficients — ?r occur in z, so x = y.
The result follows as / is anti-symmetric in x, y. We now deal with cases 2, 3,
4.
Case 2. By permuting the i;, we can assume the 2 occurrences come from
Xi, X2 and by changing an even number of signs, we are reduced to
Si = -H + + + +-H
Qj= — — — — — — corresponding to w in Lemma 6.2
Ci = -t--|--|- + + + +-|- corresponding to v
7i= — — -I- + + -I-+-I- corresponding to 62641;.
Case 3. By similar permutations, we can assimie the four occurrences to be
in Xi, X2, X3, 14 and with sign changes, we have
?i= -H-f-+-H
Qi= — — — — — — corresponding to w
Pi = -I--I--I- + + + +-I- corresponding to v
7i= — — — — -I--I-++ corresponding to 62646668^.
Case 4. We assume the 2,2,4 occurrences are in 2;i2;2i x^x^, XiXsX^Xi and
then we have, with sign changes,
?i=- -H-f--H-H
Oi = — — — — — — corresponding to w
Pi = — — -I- + + -I-+-I- corresponding to 65641;
7i=-f--l-— — +-I-+-I- corresponding to 66681;.
We check the identity in each of the remaining cases.

42 Chapter 6
Case 2. [v,w]= i{eie2-\ 1-eiseig), by Lemma 6.2, so
[[v,m], 65641;] = iFi62 H H 615616N2641; = iF264)(-2 + 6)w = -AeiC^V.
[ [62641;, u;],i;] = F1 +ie2)F3 +164I; = F1 + 162) F463641;+ 1641;)
= F1 + i62)Bi64v) = —21616263641; — 262641; = -46264V.
Since the weight of [11,62641;] does not occur in A"*", [v, 62641;] = 0. Thus
[[i;,u;], 62641;] + [[w;,6264i;],i;]+ [ [62641;, i;],u;] = 0.
Case 3. Here,
[ [v, w], 626465681;] = i(ei62 H + 615616N26466631;
= i(e2646668)(-4 + 4)ii; =0, and
[6264666811,11;] = 0, [6264666811,11] = 0,
so the identity holds.
Case 4. [62641;, w] = F1 + ie2)(e3 +164I;, so
[ [62641;,«;], 66681;] = F1 + ie2)(e3 +164N6681;
= 6668(-46264i;) = -4626466631;,
[ [666811, «;],6264ll] =—46264666811, and [6264!!, 666811] = 0
because the weight is not in L. Thus
[ [62641;, w;], 66681;] + [[«;, 66631;], 62641;] + [ [66631;, 62641;], w] =0,
and we have shown that indeed, with n = 8, L+A"*" is a Lie algebra with invariant
inner product. This finishes the proof of Theorem 6.3. Q
The Killing form.
For any two elements x, y of a Lie algebra A, the map A —^ A given by
z ^-^ [x, [y, z] ] is hnear, so we may set {x, y)K = Tr{z i-> [x, [y, z] ]), the Killing
form of A.
Remcirks 1. The Killing form is symmetric and bilinear.
Proof. It is clearly bilinear. If we define a,P : A —>¦ A by a{a) = [x,a],
0{a) = [y,a], then {x,y)i, = Tr{a0) = Tr{0a) = (i/,x)^. Q
2. The Killing form is invariant under A, for if we define 7 : A —» A by
7(a) = [z,a], then, using the Jacobi identity, we have that taking Lie product
with [i, y] is the map a/3 — /3q and with [x, z] is the map 07 — ja. Thus
([x,y],z)K + [y, [x,z\ )k = Tr{aPj - 0a-y + fia-) - 0-ya} = Tr{a0j - 0-ya) = 0.

The exceptional groups: construction of Eg 43
3. Clearly the Killing form of an abelian Lie algebra is zero.
Lemma 6.4. The Killing form on L + A'^ is non-singular: indeed,
{x,y)K = -2'iO{x,y).
Proof. Both ( , )k and ( , ) are invariant tinder SpinA6) and L and A"*"
are irreducible representations of Spin{l6) which are not dusj to one another. We
can define a map / : A —>• A, where A = L + A''", by (x,1/)k = {fx,y) for all
2/ € A and then A splits as a sum of eigenspaces of / invariant tinder Spin A6).
Thus {a+u,b+v)K = \{a,b)+fJ,{u,v). But ( , )k and ( , ) are invariant tinder A
and A is an irreducible representation of A, for the only possible subspaces closed
tinder L are L and A"*" and they are not closed under A"*". Thus A = /i, and to
find A we calculate F162,6162)^- = Tr{z 1-* [6162, [6162,2]]) = Tr(S), say. Now,
[6162,6162] = 0, [6162,ei6r] = 2626r, [6162,626^] = -2ei6r and [6162,6re,] = 0, for
2 < r < S. So 15F162) = 0, E(ei6r) = —46i6r, l5(e2er) = —4626,, 6F,6,) = 0 Eind
Tr^E = —112. On A''' the action is Clifford multiphcation, so [6162, [6,62,u] ] =
61626162^ = —ii, so Tr^+(E) = —128 and we have TrJ^{S) — —240. Hence
A = —240 since F162,6162) = 1. []

Chapter 7
Construction of a Lie group of
type ^8
Our construction of the simple, connected, compact Lie group with Lie algebra of
tjrpe Ei constructed in Chapter 6, proceeds according to the following steps.
Step 1. Take the Lie algebra L+A"*" (over R or C), where L = L{Spin{l6)).
Step 2. Take the group of automorphisms of this Lie algebra; this is the
closed subgroup of GL{L + A""") preserving the Lie bracket.
Step 3. Take the identity component and call it G. (In fact the result of
Step 2 is Eilready connected.)
AU our constructions are invariant under SpJnA6), over R or C, so we get
a map 5pJnA6) —» Aut{L + A''") and since SpJnA6) is connected, we get a
homomorphism into G. To find the kernel, note that ejej • • • eig G Spin(l6) acts
as i* = 1 on A''". It covers —/ € S0A6), so it acts as —1 on R'® but it acts
as 1 on L. Therefore it acts as 1 on L4-A"*". This and the identity are the only
elements which act as 1 on L + A"*", so we get an embedding Spin{l6)/Z2 —» G.
Now we proceed to check that G has the required properties.
Let A be a finite dimensional algebra over R or C (for example, a Lie algebra)
and let Aut{A) be the group of automorphisms of A, that is, linear bijections
a : A -* A such that a{ab) = a{a)a{b). Then Aut{A) is a closed subgroup of
GL{A), hence a Lie group.
Definition. A iinear map d : A -* A is a derivation if d{ab) = {da)b +
ad{b).
Lemma 7.1. L[Aut(A)) is the algebra of derivaiions of A.
Note that the commutator [d, &\ is a derivation if d, d are, so the
derivations form a Lie algebra, Der[A). We have a map 9 : A —+ Der{A) defined
45

46 Chapter 7
as follows: x e A maps to dx, where dx(y) — [x,y]- One checks that d^ is
a derivation and d a map of Lie silgebras. The next lemma says that d is an
isomorphism if the Killing form is non-singular.
Lemma 7.2. If A is a Lie algebra mth non-singular Killing form then its
algebra of derivations is A.
Corolleiry 7.3. With the above constructions, L{G) = L + A"*".
Proof. Clear from Lemma 6.4, Lemma 7.2 and Lemma 7.1. Q
Proof of Lemma 7.1. First, L{Aut{A)) C Der{A). To see this, take a short
curve in Aut{A), Qj = 1 + 7* + o{t'^), starting at the identity. Thus Qt(o6) =
Q«(a)Q{(b) which gives us 7(ab) = 7(aN+a7(b), so -f € L(Aut{A)) is a derivation.
Conversely, we show that Der(A) C L{Aut{A)). ]i 6 : A-^ A isa derivation,
we have by induction 6"{ab) = E (l){S*a.){5^b). Define at : A -* A hy
at= Z ~S". Then at{ab) = E ^(<5'a)(<5'6) = (Qta)(«t6), so a, € Aut{A},
n=0 n! t,j I \ ]\
and the tangent vector to q is 6 e L{Aut{A)). Q
Proof of Lemma 7.2 (Zassenhaus). To see that d is mono, suppose x € A
and dx = 0, i.e. [x, y]=0 for all y. Then [w, [x, y] ] = 0 for all y, i.e. [w, [x, -] ]
is the zero function so Tr[w, [x, -] ] = 0, i.e. {w,x)k = 0 for all w, so i = 0 as
{ ,)k is non-singular.
To see that d is onto, let us identify dx with x and extend the Killing form
from elements x to derivations by setting F,?)k — Tr{6e : A —^ A). We check
that ([7,<5],?)if + (S, [7,?])a- = 0 so the extension is invariant. Notice that the
element 6{y) should be identified with the derivation [S,dy], since [6,dg](z} =
6{dyz) - dy{6z) = 6[y,z\ - [y,6z\ = [6y,z\ + [y,5z\ - \y,6z] = [5y,z]. Thus, on
substituting 7 for i = dx and e for z, the above invariance becomes (—6x,z)}{+
{6,[x,z])k = 0.
Lemma. If {6,y)K = 0 for all y then 6 = 0.
Proof. Invariance, (—Ei,2;)k--I-F, [x,2:])a-= 0, leads to {—Sx,z)k'=Q for
all 2; in A and since the Killing form is non-singular, we have —6x = 0 for all
X, that is, E = 0. ?
To continue our proof that d : A—<¦ Der(A) is onto, take 6 € Der{A). Then
{6,y)K is a linear function in y so has the form B;, y)K for some a: G -4, as the
Killing form is non-singular. Thus {6 — dx,y)K — 0 for all y, so by the above
Lemma 6-dx = Q and 6 = d^ for some x. (Cf. [10, p. 74].) ?
Real Forms of E^.
If this construction is performed over R and the inner product on A""" is t?iken
to be positive definite, then we have

Construction of a Lie group of type Eg 47
Proposition. The group, G, obtained is compact.
Proof. The inner product on L is positive definite, so also positive on L +
A""", and G preserves it. Thus G is a closed subgroup of 0B48) and hence
compact. []
With the same details, consider the subspace L+iA''" C C(Si(L+A"*"). This is
a real vector space with the same complexification as before. Also for all a,b € L,
u,v & A"*" we have [0,6] 6 L, [a,m] ? iA""", [m,in] = —[w.f] G L, so we have
a Lie algebra over R again, which is not isomorphic as a real Lie algebra to
L + A''". To see this, look at the signatures of the Killing forms. For the compact
version, the Killing form is negative definite, with signature —248. For the other
version, ( , )jf is negative definite on L, dimL = 120 and positive definite on
lA"*", dim(iA"'") = 128, so the signature of ( , )jf is 8. The second case is exactly
the same as if we had taken the inner product on A'*" to be negative definite in
our construction.

Chapter 8
The construction of Lie groups
of types F4, Eq, E^
In this chapter we finally finish the discussion of Theorem 6.1, beginning with the
construction of the remaining exceptional Lie groups and then the representations.
We also complete our treatment of the real forms of E^.
Step 1. The usual map Spin[\2) x Spin{A) —> 5jnnA6) leads to
Spin{A) —> Spin{l2) x Spin{A) —» 5pinA6) —> E^
SU{2) —» U{2) -^ 50D).
Step 2. Take the centraliser of the image of this SUB) in Eg.
Step 3. Take the identity component and call it Ej.
Note. If we work with the compact versions of the groups, we get Ej as
a closed subgroup of Eg, hence a compact group. If we work with the complex
versions of the subgroups inside the complex version of Eg, we get the complex
version of Ej.
Step 4. The map Spin{lO) x Spin(G) —> Spin(lG) gives
SpinF) —> Spin(\6) —> Eg
' \
SUC) -^ UC) -^ 50F).
Step 5. Take the centraliser of the image of this SU{3) in Eg.
Step 6. Take the identity component and call it Eg. We now have
Eg C Ej C Eg.
49

50 Chapter 8
Step 7. The map Spin{9) x Spin{7) —> Spin{\G) gives
Gj <—> Spin{7) —» 5pinA6) —>¦ Eg.
Step 8. Take the centraliser of d in Eg.
Step 9. Take the identity component and ceiII it F4.
Remark. The embedding G2 C SpinG) depends on the choice of a point
in 5^ C A. In Theorem 5.5, we had SUC) C G2 embedded using the first six
coordinates. On the other hand, if the embedding SU{3) C G2 is by the last six
coordinates, we will get Fi C Ee- Then we have
SUB) C SUC) C G2 1 ^
FiCEeCEj J '¦
Identifications of tlie subgroups H.
We now proceed to check the subgroups H as in Theorem 6.1.
1. Take F4 first.
From the sequence
G2 •—> Spin{9) X Spin{7) —> Spin{l6) —> Eg,
it is clear that 5pm(9) centralises the whole of Spin{7) and hence it centralises
G2 C Spin{7). One can check that Spin{9) —> Eg is mono, so that F4 has a
subgroup Spin[9).
2. Next take ?7-
We have the diagram
Spin(l2) X Spin{4) —> Spin{l6) —> Eg
{Spin{l2) X S^) X SU{2).
The embedding of the two copies of 5^ is as ker A"*" and ker A~ ; we take the
embedding of SUB) as kerA+. We see that Spin(l2)xS^ centralises SUB) =
S^ = Sp(l) and hence that we have a map Spin(l2) x 5^ —> Ej. To find the kernel
16
of this, consider Spin(l6) —> Eg which has kernel Z2 generated by H Sj; the
i=\
kernel of Spin{l2)xSpin(A) —> Spin(\6) is Z2 generated by (-1,-1). Therefore
SpinA2) X Spin{4) —^ Eg has kernel generated by (ei...ei2, 613... eje) and
(—1,-1). But 613... eje lies in ker A' and not in kerA+ whereas — 613 ... eie
is in kerA""" and not in ker A". Hence only the Z2 generated by (ei...ei2,
613... eie) lies in 5pinA2) x 5^ c Ej. Hence Spin(l2) x 5^/Z2 —> ?7 is mono.

The construction of Lie groups of types F4, Eg, Ej 51
3. Finally, take Eg.
Consider the diagrsims
Spin{lO) X SpinF) —* Spin{l6) —> Eg
and
{z,g) S'xSU{3) > SpinF)
1
{z\g) 5' X SU{3) —» U{3) —^ 50F)
where the maps are the obvious ones.
We have Spin{lO) x 5' x SU{3) -^ Spin{l6) -^ Eg where Spin{W) x 5'
centralises the (image of) SUC) in Eg. It remains to check the kernel of
Spin{lO) X 5' —> Eg. The kernel of Spin{16) —> Eg is Z2 generated by
6162...ei6 and the kernel of Spin{lO) x Spin{6) —> 5pinA6) is Z2 generated
by ( — 1,-1). Thus kerEp2nA0) x SpinF) —> Eg) has four elements,
generated by (ei... eio, en ... eie) and (—1,-1), so this kernel is Z4, with all
four elements in 5'. (To see this, note that 5' is the image of t >-+ (cosi +
(sini)eiiei2)(cosi + (sini)ei3ei4J(cosi -I- (sini)ei5ei6J and for t = 7r/2,7r,37r/2
this image goes through euen ¦ ¦ ¦ eie, —1, and — 611612 ... 616 respectively,
corresponding to the points i, — 1 and — i of 5' in C.) Thus the subgroup of Ee is
Spin{\Q) X S^/Zi as claimed. This completes the check of subgroups mentioned
in Theorem 6.1.
Identification of L{G) and L{G)/L{H).
Our tactics are as before. We know L(Eg) = L(Spin(l6)) + A"*" as a
representation of SpinA6) and we have a subgroup H x. K mapping into Spin(l6).
We wish to determine the centraliser G oi K in Eg, so we write L{Eg) as a
representation of H y. K, where K is one of the groups SU{2) C SU{3) C G2, and
take the part fixed under K. This is [j(G) and we regard it as a representation
of H.
Case 1. G = F4. By restricting from Spin{16) to Spin{9) x SpinG) we
have LEpinA6))i-> LEpin(9)) + LEjnnG))+A^(giA) and A+i-+A(g>A.
By restriction from SpinG) to G2, we have L(Spin{7)) = L{G2) + a where
a is the basic 7-dimensional representation of Gi- (The roots of Spin{7) which
are not roots of G2 are ± (ij -f- Xj) = ^i* and 0, once.) Under the restriction,
A' I-+ a, A I-+ 1 -f- a. Hence
L{Eg) = L(Spin{9)) + L(G2) + a -I- A^ (gi a + A (gi A + a).
The part fixed by G2 is L(Spin{9)) -f- A ® 1 which is therefore equal to L{Fi).

52 Chapter 8
Note. We now have a map F^ x G2 —* Eg, so we may ask: what is [j(Ea)
as a representation of F4 x G2 ? Well,
L{Ea) = L{Fi X G2) + A + A^ + A) (gi a
so we have
Corollary 8.1. There is a representation of F4 of degree 26 = 1 + 9 + 16
whose restriction to Spin[9) is 1 + Ag + A. This representation is real (for
compact forms of the groups) and its weights are 0 (twice), ±1,, i = 1,2,3,4,
Kill ±i2 ±13 ±14)- D
Case 2. G = Ej. Here we restrict from Spin(lG) to 5pinA2) x Spin{A)
and LEjnnA6)) >-+ LEjnnA2)) + L(Spin(A)) + A12 ® A4 and the representation
A"*" of 527in( 16) restricts to the representation A+^A^+A'^A" of 5pinA2)x
Spin{4). Now, SpinD) = 5^ x 5^, where we regard the second factor as SU{2) =
ker A"*", and under the isomorphism A], A"*", A" correspond respectively to A'®
A', A'® 1, 1®A'. Thus
L{Ea) = L(SpinA2)) + L(S^) + L(S^) + AIj(gi A' (gi A' + A+ (gi A' (gi 1 + A" (gi 1 (gi A".
The part fixed under SU{2) is LiEj) = L(Spin{U)) + L(S^) + A+ (g A'.
We now have a map Ey x. S^ —* Eg and as a representation of Ej x S^,
L(Ea) = L{E7 X S^) + {X\j (g A' + A" ® 1) ® A',
so we obtain
Corollary 8.2. Ej has a representation whose restriction to 5pinA2) x 5''
is XI2 ® A' + A" <8i 1, with degree 56, and it is symplectic for compact forms of
Et Q
Case 3. G = Eg. When we restrict from 5pin( 16) to 5jnnA0)x5p2nF),
L{Spin{16)) restricts to L{Spin{W)) + L[SpinF)) + X\o<S\l and A+ restricts
to A+(g A+ + A" (gA".
Write ^ for the identity representation of 5' = U{1). Then, on restricting
SpinF) to 5' X 5C/C) under our map 5' x SU{3) -^ Spin{6), we find that
L(Spin{6)) restricts to LE'x 5[/C))+^''(gA2+f-''(gA'. (The roots of 5pinF)
which are not roots of 5' x SU{3) are ±{xi +Xj).) By looking at weights, we
see that A+, A", A^ restrict respectively to ^^ (g 1+^"'<8 A', ^~'(g 1+^(g A^,
^^ (g A' + f-2 (g A^. Putting all this together gives
L{Ea) = L{Spin(lQ)) + L{S') + L(SUC))
+ ^''® A^ + r'® A' +A}o®^^® A' +A}o®r^® A^
+ A+ ® ^^ ® 1 + A+ ® ^-' ® A' + A" ® r' ® 1 + ^" ® ^ ® A^

The construction o/Lie groups of types F^, Ee, Ej 53
where L(Eg) is the part on which SUC) acts trivially:
L{Ee) = L{Spin{W) x 5') + (A+ ® ^^ + A" (gi T^)-
Note. We have a map Eg x SU{3) —> Eg now, so we may consider L(Eg)
as a representation of Eg x SUC). Regarded thus,
This leads to
Corollary 8.3. Ee has two representations whose restrictions to Spin{W) x
5' are respectively ^"^ + X\o (gi ^^ + A+ (gi C^ and ^* + A}o ® ^"' + A" (gi f.
These axe of degree 27 and complex conjugates of each other. Q
This completes the discussion of.Theorem 6.1. For further information on
representations of exceptional groups, see [Ad 2]. []
The Real Forms of Eg, continued (from Chapter 7).
Our data are: a complex matrix group G; an element i G G of order 2;
A C L(G), a Lie subalgebra over R such that C (g A S L{G) and the adjoint
action of x preserves A.
Example. G = the complex form of Eg; x = —I 6 Spin(lG) -^ Eg; A =
L{Spin{l6)) + A"*", the Lie algebra for the compact form of Eg. Q
We decompose A as A""" + A", where x acts as 1 on A"*", and as —1 on A~.
Example. A+ = L{Spin{16)), A" = A+. Q
To continue, C <g A+ is the Lie algebra of the centraliser of i G G and
[A+,A+] C A+, [A+,A-] C A-, [A-,A-] C A+. In L{G), form A+-t-iA-.
(For example, form LEjnnA6))+iA+). Because [A+,iA~] CiA', [iA~ ,iA~] C
-A"*" = A""", we see that A"*" + iA~ is a Lie subalgebra over R of L(G) with
complexification L{G). (For example, A+ + iA" = L(Spin(lG)) + iA+.) If the
Killing form is non-singular on L{G), there is an easy construction for closed
subgroups H C G with L(H) = A+ -I- lA', namely
H = {g eG \g preserves A+ + iA" C /y(G) }.
Example. For x, take the image of -1 G SUB) -> Spin(i) -* SpinA2) x
Spin{i) —> SpinA6) —> Eg. This gives us an element of order 2 in G, the complex
form of fi's; A = L(compact form of Eg). Now i centralises Ej, because the

54 Chapter 8
whole of SU{2) does so, and centralises SU{2). Thus L{E7 x SU{2)) C A+
whereas
L{Ei)/L{Ejy.SU{2)) = (the 56-dimensional representation of E^) ®A' C A',
where A' is the representation of SU{2). When we form our new Lie algebra
A"*" + iA' and the new Lie group H over R, the Killing form is negative definite
on A+ of dimension 136 and positive definite on iA~ of dimension 112. Thus
the Killing form has signature —24 [^ —248,8). Therefore we have a third form
of Eg. (In fact there are only three though we do not prove this here.) Q
Note. ker(?'7 X 5C/B) —> ?'8(compact)) = Z2 generated by (—6162 .. .612,
-613614615616) where —613614615616 corresponds to —1 € SU{2). Using the
theorem that any conjugacy class meets the meiximal torus, to find all possible
elements of order 2, we may look in T.
Exercise. Show that T C Spin{\%)/Zi C fis contains 2' - 1 = 255
elements of order 2, falling into two conjugacy classes (that is, two orbits
under the Weyl group of Eg) as follows:
120 in the identity component of the centrediser of Ej x SUB)/Zi2,
135 in the identity component of the centraliser of 5pinA6)/Z2.
(See [Ad 1].)

Chapter 9
The Dynkin diagrams of F4, Eq.
Ej, E^
The Dynkin diagram and roots of G2 were given in Theorem 5.5. Using the
identification of [j(G) in Chapter 8, we complete this task here by giving the
roots, simple roots and Dynkin diagrams for the remaining exceptional groups.
If two Lie groups are isomorphic, their Lie algebras are isomorphic since G i—>
L{G) is a functor. Root systems are constructed by choosing a meiximal torus
for G or a Cartan subalgebra for L{G). Any two maximal tori are conjugate
(Corollary 1.7) and any two Cartan subalgebras are conjugate. Construction of a
Dynkin diagram requires a further choice: a fundamental Weyl chamber (see the
diagrams of the classical groups in Chapter 1, Examples 1.10). The Weyl group
acts simply transitively on the Weyl chambers [1, p. 110]. Thus if two (simple) Lie
groups are isomorphic, their Dynkin diagrams will be the same. For full details
see [Ha, p. 140]. By comparing with the Dynkin diagrams in Chapter 1, we see
that no exceptional group is isomorphic to a classical group.
Take T C Spin{9) C i*4. Then the roots are
±1, ± ij, 1 < i < j < 4 : there are 24 of these, all long;
±ii, 1 < i < 4 : there are 8 of these short roots;
(These 32 roots come from Spin{9).)
^(±11 ±X2 ±X3 ± X4) : there are 16 of these short roots, all from A.
The Weyl chamber is the one containing (8, 1, 2, 3). All roots take non-zero
integer values here and exactly four take the value 1, namely
1 ^(Xi- X2 - X3 - Xi),
2 12,
3 —12 + X3,

56
Chapter 9
4 -13 + X4,
which must therefore be the simple roots. The Dynkin diagram is
12 3 4
s s t a
Since the 26-dimensional representation restricts to 1 + Ag + A on 5pin(9), its
weights are 0 (twice) and the 24 short roots.
?"8.
Take T C Spin{l6)/Zi C Eg. The roots are
±Xi±Xj : these are from 5pinA6) and there are 112 of them;
i(± I] ± X2 • • • ± Is), where there are an even number of — signs .
(There are 128 of these from A"*".)
AU roots have the same length. The Weyl chamber is taken to contain B3, 0,
1, 2, 3, 4, 5, 6). At this point each root takes non-zero integer values and exactly
8 of them are 1, so these are the simple roots:
1 ^{xi + X2 — X3 — Xi — Xs - xe — Xt — xg),
2 -12 + 3:3.
3 I2 + 3^3.
4 -13+X4,
5 —14 + X5,
6 ~Xi + xe,
7 ~xe + X7,
8 —17 -I- Xs-
The Dynkin diagram is
Ej X SUB)/Z2.
The roots of Ej are those of fig with the same coefficient of I7 and ig. Thus
the roots are
±{x-r-xi) ( 2) for SU{2),
±{xj + xg) ( 2)
±ii±Xj l<i<j<6 F0)
5(± ii ± 12 • ¦ • ± xe ± (xy + ig)) F4) (where half the signs are —).

The Dynkin diagrams of F4, Ee, Ej, Eg
57
The Weyl chamber is to contain A3, 0, 1, 2, 3, 4, 0, 1). Hence the simple roots
1
2
3
4
5
6
7
8
^ (ll +X2-
-I2 +X3,
X2 +I3,
—13 + I4,
—14 +I5,
-I5 +l6,
—17 +l8,
I7 + Is-
- I3 — I4 — I5 — Ig — I7 — Xg),
The Dynkin diagram (of Ej x SUB)/Z2) is
3
7 12 4 5 6 8
Ee X 5[/C).
The roots of Eg are those of Eg which contain xe, 17, Xg with the sjime
coefficient. Thus the roots are
±{xi-Xj) 6<i<j<8 ( 6) for SU{3)
±Xi±Xj l<i<j<5 D0)
\(±Xi±X2±Xi±Xi±Xi±(xe+x7 + Is)) C2)
(where we take half of these with — signs).
The Weyl chamber contains (8,0,1,2,3, —1,0,1) and the simple roots are
5A1 + I2 - 2:3 - I4 - I5 — ig — I7 - Xg),
-I2 +I3,
12 +13,
—13 +14,
-I4 +2^5,
i(ii - X2 ~ Xj — Xi - x^ + Xe -\- Xj + Is),
-ig + I7,
8 —17 + ig.
The Dynkin diagram (of Ee x SU{3)) is

Chapter 10
The Weyl group of E^
Theorem 10.1. The Weyl group W{Es) of Eg coincides with the group of ail
automorphisms of the root system of E^ and its order is 2''' • 3* • S'^ • 7.
Although we will not need to quote this theorem in the sequel, we will need
to rely on several matters introduced in the course of the proof. The proof will
involve a series of lemmas and to begin, consider the diagram
Spin(l6) —» Es
I
SU(8) -^ U{8) —* 50A6).
Lemma 10.2. The identity component of the centraiiser of SU(8) is 5^
(for the compact version o/fig).
Proof. Consider the action of SU{8) on L(Eg). As an 5f7(8)-modiile,
L(Spin{l6)) = L(U{8)) + A^ + A« = L{SU(8)) + 1 + A^ + A«,
A+ = 1 + A2 + A" + A8 + 1.
Thus L(centraliser of SU{8)) is of dimension 3 and if we work over C, it is
generated by u = 6162 + 6364 + • • • + eiscie, v and w, where [u,v] = 8iv,
[u,w] = —8iu), [v,w] = iu, see Lemma 6.2. We now have a compact, connected
Lie group of rank 1, of dimension 3 and wish to know whether it is 50C) or
8 ,
5 . Look at the 1-parameter subgroup with tangent vector M : 7t = H (cos t +
(sint)e2r-ie2r). At t = 7r/2, this passes through 6162...eie = 1. Change scale
to jM = 6162 H +615616. Then [^u,v] = 2iv, [ju,w] = —2w, [v,U)] = iu, so
the roots are ± 2ii and the centraiiser is 5p(l) = 5^. Q
Lemma 10.3. In W^E^) there exists an element which reverses the vector
u = 6162 + • ¦ • + eiseie G L{T) and Rxes T(SU{8)).
59

60 Chapter 10
Proof. In 5^, conjugation by j inverts the circle e'* and fixes the whole of
SU{8) since our 5^ arises as the centraHser of SU{8). []
Lemma 10.4. W{Es) is transitive on the roots.
Proof. W{Spin{l6)/Z2) is transitive on the 112 roots, ±ii±i;, of Spin(lG)
and also transitive on the 128 weights ^{± xi ± X2 ± ¦ ¦ ¦ ± xg) of A+. Thus the
roots of Eg fall into at most two orbits. But the reflection in the hyperplane
perpendicular to m = 6162 + • ¦ • + eiseie, given by Pu{v) = v — 2-
u ¦ u
¦X2 — Xz — • • ¦ — Is), so there is only one orbit. Q
Lemma 10.5. For each root d of Eg, Eg contains a subgroup oflocal type
T^ X S^ which contains T{Eg) and whose roots are ±9. The map S^ —> Eg
is mono; this subgroup and its 5^ subgroup are uniquely determined by 6 and
subgroups corresponding to different choices of 6 are conjugate. Conjugation with
a suitable element of this subgroup gives an element of W{Eg) interchanging 6,
— 9 and Bxing all weights perpendicular to ± 9.
Proof. We already know most of these parts for two pairs of roots, namely
for ±5(iiH his) by Lemma 10.2 and Lemma 10.3 and for ±(x7-i8) because
we have studied the subgroup SUB) —> Spin{A) > Spin{l6) —* E^, in
cx>ordiaates
Chapter 8 in connection with ?7, and can adjoin the remaining elements of the
torus to that. By Lemma 10.4 the result holds for all roots 9. For the uniqueness,
look at the 2-dimensional subspace of the real vector space L(Eg) on which T
acts via ±9 (after complexification). This subspace is unique and under [ , ]
generates a 3-dimensional subalgebra characterising 5''. []
2 1
By • • we will mean roots 9i and 9^ of Eg meeting at 120 .
2 I
Lemma 10.6. W{Eg) is transitive on the pairs • • and their number
is 56-240 = 2^-3-5-7.
Proof. By Lemma 10.5, we can take 9i = —X7 + xg and count how many
roots have inner product —1 with 9i. We have
±Xi+X7, j<6 A2),
±Xi-X8, i<6 A2),
^(±11 ± • • • ± le + I7 — Xg), even number of — signs C2),
making a total of 56. There are 240 roots of Eg (see Chapter 9), so 240 possibilities
for 9i. Using the Weyl group of Spin{\2), which fixes 9^, we may permute
xi,... ,Xg in any way and change an even number of signs ; in this way we get three

The Weyl group of Eg 61
orbits. If we use W(Spin{l2) x 5p(l)), we have an operation fixing ii, ... ,ig
and sending I7 1—> —is, xg >-+ —17. This interchanges the first two orbits, so
we are down to two orbits of size 24 and 32. Finally the reflection reversing
^(ii+i2H f-is) fixes 61 and sends 11+17 to 5A1-12 xe+xr-Xg).
Hence under W{Eg) there is only one orbit. []
2 1
Corollary. For each pair • • , Eg contains a unique subgroup of iocai
2 1
type T* X SUC) containing T(Eg) and with • • as its DynJcin diagram.
That is, 9i and 62 qualify as a pair 0/ simple roots for the subgroup. The
subgroups arising in this way are all conjugate and SU{3) —> Eg is mono.
Proof. By transitivity, we need concern ourselves only with
X-j — Xq Xg — Xj
• • : take the diagram SUC) -> SpinF) -> Spin(l6) -> Eg and
fill out to a meiximal torus. []
3 2 1
By a "triple • • •" we will mean a triple consisting of three roots 9i,
62, 6i of Eg such that 9i, 9^ make an angle of 120° and similarly for 62, 63, while
61, 63 are orthogonal. Below we use the same language for other configurations
of roots.
321
Lemma 10.7. W(Es) is transitive on the triples » • • and their
number is 2^ • G* • 5 • 7.
2 I XT — xe xg—Xj
Proof. By Lemma 10.6, we can take • • to be • • . Now
the roots orthogonal to —17 + xg are
±A7 + Xg),
±Xi±Xj, I <i < j < 6,
5(±ii ± • • • ±i6 ± A7 + xs)), even number of - signs,
and those which in addition have inner product —1 with —Xe+Xy are
-Xj-Xg, A),
±Xi + xe, A0),
^{±xi±---±Xi-xe-XT-xg), A6),
making 27 in all. Using the Weyl group of Spin{W), which fixes 61, 62, we can
permute ii,... , 15 and change an even number of their signs ; in this way we
get three orbits. The reflection which reverses 5A1 + • • • + is) and fixes 9i,

62 Chapter 10
^2,
5lx
sends ij + ig to ^{xi — x^ — • ¦ ¦ — x^ + Xg — xj — Xg) and —xy — Xg to
1 + • • • + Xe — X7 — Xg). So under W{Eg) we get only one orbit. Q
Corollary. Each triple • • • arises as the Dynkin diagram of a
subgroup o/iocaJ type T^ x SU{4) containing T{Ea).
Proof. The proof is as above. Q
4 3 2 1
Lemma 10.8. W{Ea) is transitive on the quadruples • • • •
and their number is 2" • 3'' • 5 • 7.
Proof. By the previous work, we may take ^3,^2,^1 as —xs+xg, —xs+xj,
—Xj+xg respectively. The roots which are orthogonal to —xs + xt, —xj+Xg
are ±ii±X; with i<j<5 and 5(±ii±-• •±i5±(x6+i7 + i8)), (even number
of — signs). Of these, the ones which have inner product —1 with —15 +X6 are
±Xi + X5, i < 4 and §(±2] ± • • • ± X4 + 15 — ig — 17 — is) (even number of
— signs), making 16 in all. Using the Weyl group of Spin(8), which fixes 9i, 62,
63, we can permute xi,... ,14 and change an even number of signs. In this way
we get two orbits. The reflection which reverses |(xi + • • • + xg) fixes 9i, 62, 63
and sends xi + X5 to |(xi — X2 — X3 — X4 + X5 — xg — X7 — Xg), so we have only
one orbit. Q
4321
Corollary. Each quadruple • • • • arises as the DynJcin
diagram o/a subgroup of local type T* x 5t/E) containing T{Eg).
Proof. As before. []
5 4 3 2 1
Lemma 10.9. W(Eg) is transitive on the 5-tuples • • • • •
and their number is 2'^ • 3'' ¦ 5' • 7.
Proof. By the previous work, we may take 64 61 as —X4+X5, —Xs+Xg,
—Xg+X7, —X7-I-Xg. The roots which are orthogonal to —X5+X6, —xg+X7, —X7+Xg
are
±Xi ±Xj, i < J < 4,
5 (± Xi ± X2 ± X3 ± X4 ± (Xs -I- Xg + X7 + Xs))
(even number of — signs).
Of these, the ones which have inner product —1 with —X4 + X5 are
±Xi + X4,
|(±Xi ± X2 ±X3 + X4 — X5 - Xg — X7 — Xg), (ten in all).

The Weyl group of Eg 63
Using the Weyl group of Spin{6), which fixes 9i,... ,6^,we get two orbits. The
reflection reversing ^(xi+-- ¦+xs) fixes 9i,...,9i and sends ii+i4to 5 (xi—
X2 — X3 + Xi — Xi — xe — Xj — xg), so we have only one orbit. Q
5 4 3 2 1.,
Corollary. Each o-tupie • • • • • arises as the Dynkin
diagram of a subgroup of local type T^ X SU F) containing T (Eg). Q
Lemma 10.10. W{Eg) is transitive on the 6-tupies
6 5 4 3 2 1
• • • • • •
and their number is 2" • 3* • 5' • 7.
Proof. As above. Q
6 5 4 3 2 1
Corollary. Each 6-tuple • • • • • • arises as the Djmkin
diagram of a subgroup of local type T^ x SUG) containing T{Ei). Q
Lemma 10.11. W{Eg) is transitive on the 7-tuples
4 3 2 1
7
and 0/these there are 2'" • 3* • 5^ • 7.
Proof. As before. Q
Corollary. Each figure J> • • • • arises as the Dynkin
diagram of a subgroup of local type T'x 5jnnA4). Q
Lemma 10.12. W{Eg) is transitive on the 8-tuples
6
8 7
and there are 2'* • 3* • 5^ • 7 0/ them.
Proof. We may take 9i,...,9j to be —17 + ig, —ig+17, ..., i2 + a:3- The
roots which are orthogonal to 9i,... ,9e are ± 5A1 H +xs). The one which

64 Chapter 10
has inner product —1 with X2 + X3 is —^{xi-\ 1-is)) so each 7-tuple extends
to a unique 8-tuple. []
Proof of Theorem 10.1. Since W(Ea} is transitive on the 8-tuples, it
follows that the group of all synunetries of the roots system is also transitive.
The effect of W{Ea) is determined by its action on these figures as the 8-tuple
is a basis for L(T)' and so both groups are simply transitive on the 8-tuples.
Theorem 10.1 follows. []
Now consider unordered figures of the form
where we have 8 roots of Eg which fall into four pairs in orthogonal 2-spaces,
each making an angle of 120°.
Theorem 10.13. Such Bgures are permuted transitiveiy by W{Eg). Each
arises as the Dynkin diagram of a subgroup H D TlEg) where H = (SU{3) X
SU{3) X SU{3) X SU{3))/Z3 x Z3 and Z{H) = Z3 x Z3. From two non-trivial
elements in Z{SUC)) = Z3 (four times) we obtain the 8 non-trivia] elements
in Z{H). Let S C W{Ea) be the subgroup preserving a figure. Then S —>
Aut{Z{H)) = GLB, F3) is an isomorphism. In particuJar, 5 maps onto E4,
tile permutations of the four SU{3) and the kernel 0/ S —> E4 = PGLB,F3)
(corresponding to —1 G GZ,B,F3)) acts as ••" '• •''~^» •'^^^ • •
Explanation. Here Z[G) is the centre of a group G. Ont{SU{3)) = Z2
acts by inverting both the centre of SU{3) and the Djmkin diagram. In other
words, the point where one root of SU{3) is 1 and the other —1 is a non-trivial
element of the centre, so if you turn the Dynkin diagram end to end, you invert
the centre and vice versa.
Proof. This will take up the rest of the chapter. We continue in the same
spirit as before.
Lemma 10.14. W[Ei) is transitive on the figures
3 2 1
• • •
and there are 2'° • 3^ • 5 • 7 0/these.
Proof. By the proof of Lemma 10.12, we may take 61 and 62 as
X7 —Ig Xi—X-j
• • . Orthogonal to both we have
±Xi±Xj, l<i<j<5 D0)

The Weyl group of Eg 65
^{±xi±-¦ ¦ ±Xi±{xe+Xt+ xg)), even number of — signs C2),
making 72 roots.
Under W{Spin{W)), which fixes 9i, 62, we have orbits
|(±Il ± • •• ±15 + (l6 +17+18)),
|(±ii ± • • • ± 15 - (le + 17 + xg)).
Using the reflection which reverses 5A1 + • • • + is) and fixes 9\, 9^, we send
ii +12 to |(ii + 12 — X3 Is)- Hence there is only one orbit under W{Ea).
u
Lemma 10.15. W{Es) is transitive on the figures
4 3 2 1
and tiiere are 2'^ • 3^ • 5^ • 7 of these.
Proof. By the previous lemma we may take S3, 9^, 9i as
I5~l4 Xj—Xg Is —17
The roots orthogonal to 9\, 9^ and of inner product —1 with S3 are
±ii + i4, 1<J<3 F),
±ii-i5, l<i<3 F),
5(±ii ±i2±i3 +14 -15 ± (i6 +17 + xa)),
even number of — signs (8),
making 20 roots.
Under W{Spin{6)), which fixes Sj, S2, ^3, we have orbits
±Xi + I4,
±ij — 15,
I (± II ± I2 ± I3 — I4 - I5 + l6 + I7 + Is),
5(±Il ±l2 ±13 — I4 - I5 — I6 — I7 — Is)-
In W{Eg) we have I4 >-+ —15, 15 i-+ —14 which fixes 9i, 9^, S3 and sends the
first two orbits into one orbit of 12 elements. By using the reflection reversing
5A1 + • • • + ig) which fixes Si, Sj, S3, we send ii +14 to 5A1 — 12 — 13 +14 —
I5 -Is) and -Xi - Z5 to ^(-ii +12+13 + 14-15+3^6+17 +is)- Thus
there is only one orbit under W(Eg). []

66 Chapter 10
Corollary. Any figure • • • • arises as the Dynkin diagram of
a subgroup o/iocai type T" X SU{3) x SU{3). Moreover, SU{3) x SU{3) -^ Es
is mono.
Proof. It is sufficient to check for —X3 + X4, —14 + xs, —xe + xr, —Xj + xg
and we can put SU{3) X SUC) -^ SpinF) x Spin{6) -* 5pinA6)/Z2 -^ Eg. Q
Lemma 10.16. For each figure • • • • the roots orthogonal to
^ii ^2, S3, 9i form a root system of type A2 x A2.
Proof. By the previous lemma, we need only consider
I4-I3 I5—14 -16+3^7 Xg—Xj
• • ¦ •
The roots orthogonal to these are
±Il±l2, D),
5(±Xi ±i2 ± C:3 +3^4 + X5) ± {xe +X7 + xs)), (8).
These split into the following subsets
±(xi +I2),
±5(-ii - i2 +X3 H l-xs),
±5(-ii - I2 -Xg)
and
±(ii -X2),
± |(-Xi + X2 +X3 + X4 +X5 - Xg — X7 — Xg),
±i(-Xi +X2 - X3 - X4 -X5 +X6 +X7 + X8).
These two subsets have the form ±a, ±/?, ±7, a + /? + 7 = 0, (which is the
root system for SU{3)) and lie in orthogonal 2-planes. Q
It is now clear that the number of ways of choosing 65,..., Sg is 12 • 2 ¦ 6 ¦ 2 =
2^ • 3^
Lemma 10.17. W{Eg) is transitive on the figures
@/which there are 2'" • 3'' • 5^ • 7) and the stabiliser has order > 48.

The Weyl group of Eg 67
Proof. To count the number, we have to divide the number of figures, namely
2* • 3^ X 2'^ • 3^ • 5^ • 7, by the number of ways of putting the numbers on, which is
2 • 4 • 6 • 8 = 2^ • 3. To prove transitivity, take any such figure and write the two
figures concerned as
<t>2
Choose a root 6i and a root 63 (there are 48 possibilities). By Lemma 10.15,
there is an element of W{Eg) taking 61,... ,6^ onto our standard roots (j>i
1^4. This standard choice of (pi,- ¦ ¦ ,(f>A, determines an orthogonal root system of
type A2 X A2. The reflections corresponding to these roots generate a subgroup
E3 X E3 of W{Eg) fixing 1^1,... ,1^4. (In fact, by the corollary to Lemma 10.15,
this E3 X E3 actually comes as the Weyl group of a subgroup of local type
T" X SU{3) X SU{3).) Now, 9^ and 9e must go to a pair of roots of one A2-
system, but we do not know which. However, by using the E3 of that A2-system,
we can make sure that 9^, 9g go to our preferred pair of simple roots, but we do
not know which one 9^ goes to or which one 9g goes to. Similarly, 9j and 9^ go
to a pair of roots of the other A2-system and by using the E3 of that A2-system,
we can ensure that ^7, 9g go to our preferred pair of simple roots in some order.
Thus W(Eg) acts transitively on these figures and we can carry one figure into
any other in > 48 ways. Q
Lemma 10.18. Each figure
as the Dynkin diagram o/a subgroup of local type SU{3)^.
Proof. The diagram • • corresponds to a unique SU{3) subgroup and
any two of the four subgroups commute since • • • • corresponds to a
unique SUC) x SUC). Thus we have a map SUB) x SUC) x SUC) x SU{3) -^
Eg. D
Continuing with the proof of Theorem 10.13, to show that a subgroup Z3 x Z3
maps to the identity we can use the coordinates from Lemma 10.16. The centres
of the four A2 work out as follows [1, Proposition 5.3]. We are to solve 9r{t) s
0 mod 1 for t GT, 1 < r < 8, and find that the non-zero solutions are given by
the following:
±i@,0,1,1,1,0,0,0), ±^@,0,0,0,0,1,1,1),
±i@,0,1,1,1,1,1,1), ±^@,0,1,1,1,-1,-1,-1). (All coordinates mod 1.)
The kernel of SU{3)^ —> Eg must be contained in the centre. The centre
of SU{3)'^ is Z3 X Z3 X Z3 X Z3 and at least Z3 X Z3 of it survives because

68 Chapter 10
we have seen that the map Sf/C)^ —¦ E^ is mono. So the subgroup of Eg is
exactly SU{3)'^/7i3 x Z3. Each map S?7C) —> Eg is mono, so from the two
non-trivial elements in the centre of each SU{3) we get two non-trivial elements
in the centre Z[H) and the pairs we get from distinct SUC) are disjoint as each
map SU{3)^ —> ?8 is mono. So, we obtain the 8 non-trivial elements of Z{H).
Lemma 10.19. Let S C W(Es) be the subgroup preserving a figure. Then
there is a monomorphism S —* Aut{Z{H)) = GLB,F3).
Proof. Suppose w e S C W{Es) maps to 1. If w preserves the four pairs
in Z{H), it must preserve the four SJ7C); if it keeps each pair in Z{H) fixed
rather than reversing it, it must fix each • • rather than reversing it, so it
fixes 61,... ,6g. But these form a basis, so it fixes L{T)' and w = 1. Q
Lemma 10.20. S -> Aut{Z{H)) is epi.
Proof 1. By Lemma 10.17, |S| > 48, so by Lemma 10.19, the image of S
has order > 48. But |GLB,F3)| = 48, so the image is GLB,F3).
Proof 2. By referring to the proof of Lemma 10.17, we see that 5 allows us
to take any non-zero a G Z{H} onto any non-zero a' e Z[H) and similarly any
b^O,±ae Z{H) onto a"^0,±a'. Q
That finishes the proof of Theorem 10.13. Q

Chapter 11
Representations of Eq and Ej
We now deduce some corollaries for the representations of Ey and Eg from the
work of Chapter 10. In Chapters 12 and 13 these will serve as gmdelines for
constructing Ej and Eg and their representations independently of Chapter 8.
Theorem 11.1. The group ?7 contains a subgroup SU{&)/{±1} ajjd on
restriction to this group, ?'(?'7) 1—> L{SU{Si)) + A* and the E6-dimensionaJ
representation 0/ Ej) 1—> A^ + A®.
Proof. Ey is the identity component of the centraliser of SU{2) C Es,
corresponding to ±(—I7 + I8). We can equally well use SU{2)CEg corresponding
to ±^{xi+X2-\ l-ig) and this 5J7B) is the centraliser of 5J7(8), (Lemma 10.2),
so we get SU{8) —>¦ Ey. We need to know the kernel of this map and this comes
down to seeing if the point 6162 ... eje G 5pinA6) lies in the image of SU{8) or
not. We have the diagram
SU{8)
and the 1-parameter subgroup
n (cost-l- (sint)e2r-ie2r) n (cost - (sint)e2r-ie2r)
is contained in the torus of the lift of SU{8). By taking t = 7r/2, we see that
6162 .. .ei6 lies in the lift of SU{8). Since it covers the element —1 G S0( 16), it
corresponds to the matrix -1 G 5f/(8), so we get SU{&}/{± 1} -> ?7. Q
Now take L{Es) as a representation of Ej x SU{2) and restrict further to
SU{8) X SU{2). Then, as an SU{8) x 5J7B)-module,
L{Es) = L{SU{8)) + \* + 2(A' + A«) + 3
69

70 Chapter 11
where the " arises from L{SU{2)).
CoroUciry 11.2. The 56-dimensional representation, A^ + A®, of E^ is
irreducible.
Proof. On restricting to 5pinA2)xS'' we get the two irreducible summands
of degree 24 and 32, but on restricting to SU{8) we get two irreducible summands
of degree 28. ?
Corollary 11.3. The action of the sumraand A* C L{Ey) on both A^ and
A® in the 56-dimensionciJ representation is non-zero.
Proof 1. If A* acted as 0 on A^, then A^ would be a representation of
LlEy); this is a contradiction since the 56-dimensional representation is
irreducible. The same argument works for A®.
Proof 2. Let Va, Vg e L{Es) be non-zero eigenvectors for T corresponding
to the roots q, P. Then [va,Vg] is an eigenvector for T with weight a + 0.
Thus [va.u^] = 0 unless a + P = 0 or is a root of E^. To complete the proof
we use
Lemma. If a + 0 is a root of Eg, then [va,vg] ^ 0.
Proof. If a + P is a root, then the angle between q and P is 120°
\ • • ) so without loss of generality, it is suflScient to check the lemma for a
subgroup SU{3). For classical groups the lemma is trivial. If a + /? = 0, we use
the transitivity of W[Ea) on roots to reduce to the case of SU{2). []
Theorem 11.4. The group Eg contains a subgroup isomorphic to
SU{3) X SU{Z) X SU{Z)/2,3 in such a way that
LiEe) H— L{SUCf) + X\1)X\2)X'{3) + A'A)A'B)A'C).
One of the 27-dimensioniiJ representations of Eg maps to
A'A)A2B) + An2)A'C) + AH3)A'A),
whilst the other maps to
A'A)A'B) + A^B)AiC) + A^C)A'A),
where A(i) denotes the usual space on which the i-th copy of SU{3} acts.
Proof. We defined Eg as the identity component of the centraliser of SU{Z)
in ?^8 and we have maps
SU{3Y ~* Es, and

Representations of Ee and Ej 71
SU{3f ->• Ee, (sending the last to the centralised SU{2)).
The latter has kernel Z3.
We regard L{Ei) as a representation of SU{3)^. Now, L{Ei) has 240 roots
and L{SU{3y) has 24. Each of the 240 - 24 = 216 roots which are not roots
of L{SU{Z)^) must restrict to a sum of weights of the four copies of SU{3}. But
SU{3} has only one sort of weight shorter than a root, namely ±ii, ±12, ±13,
and the weights of A', A^. The length of each of these is -i times the length of
a root. We get one of these roots in three of the four cases, 0 in the fourth.
One can check this directly or as follows. If a root of ?^8 restricts to 0 on two
copies of SU{3), then it is orthogonal to • • • • , so by Lemma 10.16,
it is a root of SU{3) x SU{2). Thus, apart from the roots of SJ7C)'', a root of E^
restricts non-trivially on at le?ist three copies of SU{3}. The roots of ?^6 -which
are not roots of SU{3)^ give us 78 ~ 24 = 54 roots of Eg which are not roots of
SU{3)^ but are 0 on the last SU{3) and we have four copies of SU{3), so we get
4 ¦ 54 = 216 roots which are non-zero on three, zero on one. There are none left to
be non-zero on all four copies. We conclude that these 216 roots are the weights
of a representation of 5f/C)'' which is a sum of terms A^'(ii)A^^(i2)A^^(i3). Thus
L{Es) = LiSUiS)") + sum of terms A^'(ii)A"^(i2)A^^(i3),
where Cj = 1 or 2 and ij, 12, 13 are 3 of 1,2,3,4 and each term is of degree
27. We get two such terms not involving factor 1 (respectively 2,3,4). Choose
non-trivial elements Zi, Z2, Z3 in the centres of the first three copies of SU{3)
so that {zi,Z2,Z3) I-* I E Eg and choose isomorphisms with the standard SU{3)
so that Zi, Z2, Z3 correspond to some element ul. Then any element of our
symmetry group S (see Theorem 10.13) which preserves the fourth SU{3} and
permutes the first three SU{3) must either fix or invert {^1,22. ¦23}, that is, must
either permute the first three SU{3) or induce the outer automorphism • •
on all of them. Thus S must contain a E3 permuting the first three copies of
SU{3) preserving their isomorphism with the standard SU{3) while fixing the
last SU{3), or reversing it according to the sign of the permutation.
It is now clear that, of the representations A^'A)A^^B)A^'C), the only ones
which are trivial on the kernel of SU{3)'^ —>¦ E^ are
A'A)A'B)A'C) and A'A)A'B)A'C).
The subgroup of S which permutes the last three SU{3) fixes A'A)A'B)A'C)
and its 8 cosets give the 8 representations we want.
In its guise as GjLB,F3), S contains (o_'5) which fixes one line, reverses one
line and interchanges the remaining two. Thus S contains an element of order
2 fixing the first SU{3), reversing the second and interchanging copies three and

72 Chapter 11
four. Identify copies three and four under this symmetry. Then we have two terms
A'A)A2B)A'D) and X^{1)\^{2)X'{4). Permuting the first three copies of 5i7C)
gives A'B)A2C)A'D), A'C)A2A)A'D), A2B)A'C)A2D) and X^{3)X'{l}X^{'i).
The coefficient of A ^D) is one 27-dimensional representation and the coefficient
of A2D) the other. Q
Corollary 11.5. The 27-dimensionaJ representations of Eg are irreducible.
Proof. On Spin(lO) x S' there are irreducible summands of degree 1, 10,
16 and on 5J7C)' there are summands of degree 9, 9, 9. Q
Corollary 11.6. The action of A'A)A'B)A'C) C L{Ee) on each of the
summands A'A)A'B), ..., of each 27-dimensional representation is non-trivial
and similarly for X^{1}X^{2)X\3).
Proof 1. See Proof 2 for Ey.
Proof 2. (Cf. Proof 1 for Et, Corollary 11.3.) Suppose that the action of
both
A'A)A'B)A'C) and X'^{1)X'^{2)X'^{3) on all the three summands A'A)A2B),
A'B)A^C), A'C)A^A) (of one representation) is zero. Then the three summands
would be representations of LlEg), contradiction. Therefore there is a non-zero
one, say the action of A'A)A'B)A'C) on A'A)A^B). Then by cyclic permutation
of A'A)A'B)A'C) the action on X\2)X'^C) and X\3)X'^{1) is also non-zero.
The representations are dual, hence A'A)A^B)A'C) acts non-trivially on one and
hence aU of A2A)A'B), ... . Apply -1 ? 5 and see that A2A)A2B)A2C) acts
non-triviaUy on all summands. By duality, it acts non-trivially on all summands
of the second representation. []

Chapter 12
Direct construction of Ej
The object of this chapter is to give an explicit construction for Et, its 56-
dimensional representation, without assuming Chapter 8 and show that what we
construct is indeed isomorphic to what we have already seen. We will exhibit ?7
as a group of linear maps in 56-dimensional space. In Theorem 14.18 we will see
that the 26-dimensional representation of F4 is a subspace of a Jordan algebra.
The 56-dimensional representation W of Ej cannot have an invariant algebra
structure for the simple reason that the non-trivial element in the centre of Ej,
say il e SU{8)/{±1}, acts as —1 on W, therefore as 1 on W ® W; therefore
any invariant map f : W i^W —>^ W is identically zero. We will reveal all the
algebraic structure that seems to be available, but the reader may feel that it
is no better than it has to be. (For structures on W, see [All] and [Bro].) The
construction will be in terms of exterior algebra.
To begin with, suppose given two dual vector spaces V, V over C. We
wish to preserve the symmetry between V, V so we lay down that V" is
to be identified with V in such a way that {v,v'} = {v',v}. Now introduce
the exterior powers A'(V), \^{V'). These are the components of the exterior
algebras A{V) and A{V'j, so we have product maps X'{V) ® X^{V) — A*+^(y)
and X\V')iS)X'{V') -^ X'+^{V'). The spaces X'{V} and X'[v') are dual in such
a way that
{vlV2...v'i, ¦WiW2...Wi)= Z eWW,W^(l))(«2.W»B))--- K',W^(i))
and this formula is self dual. The space V ® V* acts on V via (u ® v'}w =
u(i;*,u;). This identifies ViSiV with Hom(y, V) = L(GL(y)), the Lie algebra
of the group GL{V}. Dually V* ® V acts on V and this identifies V iS) V
with Homlv',V') = L{GL{V'}), the Lie algebra of the group GL{V'). We
normally identify GL{V) with GL{V') by {gv',v) = {v',g~^v). This has the
effect of making the pairing invariant: {gv',gv) = {v',g'^gv} = {v',v). If we
want to proceed infinitesimally with the Lie algebra, the statement that the Lie
algebra preserves the pairing is {¦yv',v) + {v'j^yv) = 0. Accordingly, we identify
73

74 Chapter 12
i; ® w' e L{GL{V)) with -w' ® i; e V* ® V = L{GL{V'}}. The identification
preserves the Lie bracket and the formula is self dual.
Next we make explicit the action of L{GL{V)) on X'{V). If we proceed
globally, then an element geGL{V) acts on A'(V) as follows:
g{viV2 ..¦Vi) = {gvi){gv2} ¦ ¦ ¦ {gvi).
However, infinitesimally, we want to see the action of the Lie algebra, so we must
put g = l + jt + O(t') and let i —> 0, retaining the terms of first order in t. We
get
t
7(lllll2 ... Hi) = E V1V2 . . ¦ 11,-1 G1'r)l'r+l ¦¦¦Vi-
Let the dimension of V be n and choose an isomorphism A"(V) = C, and
dually A"(V") = C. The symmetry group preserving these isomorphisms is
SL{V), identified with SL{V') as above. All subsequent constructions will be
invariant under this symmetry group. Its Lie algebra L{SL{V)) = L{SL{V'))
is the set of matrices of trace 0, which may be identified with the kernel of
{,):V®V -*C and ( , ) : V ® V-* C.
Consider now the product map \'{V) ® X'{V} —> \"{V) = C, assuming
i+j = n. This map establishes A'(V) and A-'(V) as dual vector spaces. Therefore
we identify \'{V) with {X'{V))', which is already identified with A^(V*). It is
very important to maintain the symmetry between V and V : if we apply the
same considerations with V replaced by V*, we see that ^'{V'} can be identified
with A'(V") = A'(V). This identification only agrees with the previous one up
to a sign (—1)'-'. In the applications, ij will be even, so the sign is +. To prove
the assertion about the sign, choose a basis ej,... ,e„ of determinant 1, that is,
the isomorphism A"(V) = C carries 6162... e„ to 1. Let ej,...,e* be the dual
basis in V*; then ej,..., e* is also a basis of determinant 1 and our situation is
symmetrical. In A'(V) and A-'{V) consider the elements e,, ...en and e,, ... 63^.
where 1 < t-j < • • • < t-j < n and 1 < Si < • • • < Sj < n. The product
(e,, ... eri)(e3, ... e,.) is 0 only if any r is equal to an s and there remains a
unique choice of sj,... ,Sj for which the r's and s's are all distinct. Then the
product is e = ± 1, where e is the signature of the permutation (t-j ... r^Si... Sj).
This says that the element e^ ... e^ G A'(V) is to be identified with eej^ ... ej. 6
A-'(V'). Replacing V by V', this element in turn has to be identified with
erje,, ... e,; where 77 is the signature of the permutation (sj... SjTi ... r^). Of
course ?77 = (—1)*-'.
Construction of L{Ej) and its 56-dimension?il representation.
We are now ready to construct candidates for L{Ej) and its 56-dimensional
representation.

Direct construction of ?7 75
Definition. Put A = L ® \* where L = C ® L{SU{8)) = L{SL{V)) =
L{SL{V')) and A" = \^{V) = \\V') where dimV = 8. Put W = X^ + A'*,
where A' = \\V) = \^{V').
We wish to see A acting as a Lie algebra of linear maps W —t W. The
components of this action are
L ® A' -^ A^ the usual action of L = L{SL[V)) on X^[V),
L®\^'-^ A'', the usual action of L on A^(V"),
and
A*®A2 =X\V)®\\V) -^X^[V) = X^{V'),
where ip is multiplication in A(V);
A" ® A^' = \\V') ® A2(V) -^ \^[V) = \^[V).
Theorem 12.1. These maps embed A as a Lie algebra of maps from W -+
W and this action coincides up to isomorphism with the action 0/ L{Ej) on its
56-dimensionaJ representation.
Proof. We have maps L -* Hom(A^A'), L -^ Hom{A'',A^*), A" -+
Hom{\'^, A'*), A"* —> Hom{>?',\^) which are all non-zero irreducible
representations of L[SL{y)), so mono (by Schur's Lemma). Thus A embeds as stated. To
check the action, choose a positive-definite Hermitian form on V compatible with
A"(V) = C. This gives a compact subgroup SUiV) C SL[V) which we identify
with the explicit 5J7(8) —> ?7. Then there is an isomorphism A^ -|- A'' —> W
(the space of the 56-dimensional representation, Corollary 8.2 and Theorem 11.1)
preserving the action of SJ7(8). More explicitly, the action of L = jL(SJ7(8)) on
A' -I- A'* corresponds to the action of L{SU{S)) '—> LiEj) on W. Similarly,
there exists an isomorphism L -I- A"* —+ L{Ej) which is the identity on L and on
A"* preserves the action of SU{S). The action of L{Et) on W must correspond
under this isomorphism to a sum of components; we want to know the components
involving A*. The action of A* must be
A*®A'^A^ A*®A'-f^A'-, A*®A^•i^A^ A^^A^'^A^',
where c,c! G C, as these are irreducible representations of SU{8). We can
change A"* -+ LiEj), A^ —>• W, A^' —>• W by multiplying by scalars 7, Q,
P, respectively. This replaces c by {a'y/P)c,c! by {l3-(/a)d and by choosing q,
0, 7 we can arrange that c = d = 1, so the action agrees up to isomorphism
with the action of L{Ej) on W. This finishes the proof. Q
However, to show that our present construction can be used to construct Ej
ab initio, we give a direct proof that the vector space of maps Hom{W, W) is
closed under taking commutators.

76 Chapter 12
We begin by giving some brackets in A and checking that they agree with
those in Hom{W, W). Some of this will work if dim V = 4m, so set dim V = Am,
A= L+A^ and on A put a symmetric form ( , ) with the following components.
If ;C,y e I are matrices, {\,Y) = Tt{XY). Make I, A^"- orthogonal and on
A^"", choose A^™ ® A^^-^A"*"" = C. Next, we construct a skew-symmetric
bihnear product [,]:(? + A'"") ® (i + A^"") -> i + A'"". Well, the components
jL ® jL —>• L and L (8) A^"" —> A^'" are respectively the usual Lie product and
the action of L on A*". We define [ , ] : A^"" (g) A^'" -> i by the equation
{a,\x,y\) = ([a,2],J/), a ^ L, x,y e A^™. This characterises [ , ] as ( , ) is
non-singular on L.
We check that [ , ] is skew-symmetric. For aU a e L, x,y e A^"" we have
(a, [^,y]) + (a, [y,x]) = ([o;X],y) + ([a,y],x) = ([a,x],y} + (i, [a,y]} = 0,
because the bilinear form ( , ) on A^"* is invariant under L. All this is strictly
comparable with what we did in constructing E^. We proceed to give an explicit
formula for the bracket A'"" ® A'"" -> L, cf Lemma 6.2.
Lemma 12.2. Let {ei,...,e4m} be a basis 0/determinant 1 in V. Then,
(i) [er,e,, ...er,„, e„e„...esj„] =0 if two or more r's are s's,
(ii) [6162... e2m, e2me2m+l •••e4m-l] = e2m^lm>
2m 4Tn
(iii) [eie2...e2m, e2m+i...e4ml =5(Eere;- E e,e;).
Proof, (i) Let aeL{SL(V)), i = e,) .. .e,,^, y = e,^...e,^^ and
suppose two of the r's are equal to 5's, say r', r". Then [a,x] is a sum of terms
each of which is a multiple either of fir' or e,", so all of them annihilate y and
([a,x],y) = 0. This holds for all a, so [x,y] = 0.
(ii) Let a e i(Si(V)), x = ei...e2m, y = eim---^im-i- Then [a,x]
contains terms each of which is divisible by 62^ and so annihilates e2m, except
for the term ej... e2m-i(ffle2m)- Thus
([1. ^1. J/) = ei • • ¦ e2m-i(ae2m)e2m • • • 64^-1 = el„{ae2m).
This has the form Tr{ab) where 6 = e2m^lm-
Let aeL{SL{V)), i = ei...e2m, J/= 62^+1 ••• e4m- Then
[a,x\ = Yl ei ...(aer)...e2m, and
l<r<2m
([a, 2;], t/) = Y, ei • ¦ • (fflCr) . . . e2me2m+l . . ¦ e4m
1 *'_*"»-*.
m
l<r<2m
= E e;{aer).
l<r<2m

Direct construction of Ej 77
This has the form Tr(^ab) for 6 = A/ + X! ^rK- Here A is a scalar which we
l<r<2m
have to choose so as to get a matrix 6 of trace 0. Taking A = —i will do. [1
For a general value of dimV, there is no reason why this bracket should
satisfy the Jacobi identity. However, we are studying the case dimV = 8. We
must check, without relying on the previous chapter, that the subspace L 4- A* C
Hom(A' + A'', A^ + A^*) is closed under the formation of commutators a.b — 6a
and these commutators agree with the formula we have given in L + A*. That is,
we wish to check that x{yz)—y{xz) = [x,y]z for all x,yeL + \^, zeA^+A^*.
When x,y e L the result is true: the action of L on A^ + A'' is a Lie action.
Next, consider the case i G L, y G A*. Here the result is also true because the
map A* ® (A' + A^*) —> A^ + A^" is invariant under L. Ebcplicitly, this means
x{yz) = [x,y]z + y{xz} (or x e L, y e A"*, z e A' + A^'. The case y e L, a; G A*
follows because the result is skew-symmetric in x,y. So it remains to consider
the case x,y G A*. Because of the symmetry between V and V, we may take
z G A' : the other case is the same. We now have a map
/ : A^ ® A" ® A^-* A^
which is invariant under L{SL{V)), namely
I ® 2/ ® z !-> x{yz) — y{xz) — [x, y]z.
We wish to prove this is zero. By the standard operating procedure, we can
assume x = eae^eced, y = e/ej,e/,ei, z = 6^6^, where a, 6,..., A; G {1,..., 8} and
a,...,d are distinct, f,...,i are distinct, and j, k are distinct. The matrix
diag{(^i,... Xs) acts on i®2/®z by multiplication with CaCi> ••-Ct- Therefore,
it acts on f{x ® y ® 2) in the same way. The result is 0 unless six of {1, ..., 8}
occur once among a,... ,fc and two, say, I, m occur twice; then the outcome
will be a multiple of ejem- The action of N{J') allows us to permute the basis
vectors in any way provided we alter the sign of one basis vector, if necessary. We
then have the following cases to check.
Case (a) 1 = 61626364, t/= 65666768, 2 = 6364,6468 or 6768,
Case (b) X = 61626364, y = 64656667, z = 6368 or 6768,
Case (c) X = 61626364, y = 63646566, z = 6768.
This completes the check that the subspace L + \* is closed under commutators
and the commutators are given by Lemma 12.2. But commutators always satisfy
the Jacobi identity, so A = L + A* is a Lie algebra, and we are finished with the
discussion of Theorem 12.1. []
Addendum. The inner product on A is invariant and the Killing form is
36(, ).

78 Chapter 12
Proof. The inner product is invariant under L by construction. Secondly,
take I e L, y,z E X^. Then {x,[y,z]) = {[x,y],z) by the definition of the Lie
bracket. This equation gives {ly,x],z) + {x,[y,z]) = 0, which shows that the
form is invariant under y G A*. For the Killing form, we argue, as we did for
L(Ei), that both the Killing form ( , )k and our chosen inner product ( , ) are
invariant under A = L + \* and A is irreducible, so we must have { , )k = c{ ,)
for some scalar c. But on L{SU{8)) the trace of z t-y \x,{y,z\] is 16Tr{xy) and
the contribution of A'' is given as follows: for x — eie^Riej, e-iejejei;, eje^e^ei,
[fijej, [1,6265)] is respectively 0,616^6^64,0 and so Trv(ei65 • e2ej) = 20. []
We go on to define further structure maps on W. Firstly, W is symplectic,
so it has to carry a skew-symmetric bilinear form. Secondly, let us point out
a principle that covers several known cases. (Compare [Al].) Suppose W is a
faithful representation of a Lie group G, e.g. Aut[A) on A for some Jordan
algebra A. Then we have an embedding i : L{G) —> HoTn{W,W), for example,
Der{A) — HomiA,A). We also have Hom(W,W) ^W®W'. If W is semi-
simple, we expect a G-map from W® W back to L{G) so that the composite
is 1:
L{G) -^ HomiW,W)^W'®W -^ L{G).
L[G) will usually have an invariant product and W ® W certainly does, so we
can construct a map back by transposing i.
Apply this procedure to W = k^ + A^*. First put a skew-symmetric bilinear
form on W with the following components
a2®a'^c, a^®a'--Hc, A^'SA^^C, a^-®a^*^c.
Write this form as ( , ). Then on W ® W we have {w\ ® ^2,^3 ® W4) =
(w\,Wi){w2,'Wi). This is symmetric and does not change sign if you interchange
V and v.
Lemma 12.3. There is a unique i : A-^W ®W such that the composite
A®W^W®W®W '^' W is a®wy—^[a,w].
Proof. The action of A on W is a map Ai® W —>¦ W which gives an
injection A —>• Hom{W, W) = W (S) W' =W®W. The proof now follows from
the fact that ( , ) is non-singular. Q
We now construct a symmetric map W ® W —> A with the following
components :
A2®A2^:^A^ A'-®A'-^^A*- = A*
and
A^® A^* -^V®V JUL,

Direct construction of Ej 79
where the first map is
U1U2® V'^V] I -* (llJ,Ui)U2®l'2- {Vi,U2)Ui®V^- (llj, Ul)U2 ® fj + (llj, U2)Ul ® tlj,
and 7r(i) = x- lTr{x)I, after using V®V' = Hom{V, V).
Theorem 12.4. The form {,), the product o and the map i are invariant
under A. The map i maps into the symmetric tensors and the diagram
/ 12 >yl
commutes. For alJ a G A, y,z & W,
(a, 2/oz)^ = (la, y®/)^®^ = ([a,2/1, z)w
Proof. Allour constructions are invariant under L. To prove that the form is
invariant under A*, we wish to check that ([i, y\, z)-\-{y, [x,z\) = 0 for all x G A"*,
y,z &W = )^ + A^'. If y, z are in opposite summands, then both products are
0, so we can assume that y, z are in the same summand. Suppose y,z E A^.
Then the definitions are {[x, y], z) = —{xy)z G A* = C and {y, [x, z]) = y{xz) G
A* = C. These add up to 0 since xy = yx. If y,z E A'*, the proof is similar:
interchange V, V. It is clear that the map i is invariant under A since all its
ingredients are. We now check the relation between {ix,iy)w^w and (x,?/)^.
Since A is an irreducible representation of A, we must have {ix,iy)wisw =
c(i,2/)^ for some scalar c and tp determine c, we check on a pair of elements.
The operation [eiej, ] carries 626^ G A^ to eje^ for 3 < t- < 8 and 6,6, to 0
otherwise; eje* to —e^K ^°^ 3 < r < 8 and e'e^ to 0 otherwise. So we have
Similarly
8 8
r=3 r=3
8 8
r=3 r=3
SO (i(e2ej),i(eie5)) = 12 = c.
To check that (i, y o z)a = ([x,y], z), consider first the case x e L, y
z G A'*. Then, the second map in the definition of y o z does not affect the
pairing with x and for y = ¦U1U2, z = dJdJ we have
{x,yoz)
= {vlUi){vlXU2) - {vlU2){vlXUi) - {vi,Ui){vl,XU2) + {vlU2){vl ,XUi)
= {vlvlui{xu2) + {xui}u2) = {vlv'2,x{uiU2)} = ([x,y],z).

80 Chapter 12
Consider next the case x & L, y e \^', z e \^. Here
(i,y o z) = {x,z oy) = ([2;, z],y), by the above
= —(z,[x,y]), by invariance under L
= ([3:,y],z) ^s ( , ) is skew symmetric.
Note that this working established that (lx,y], z) is symmetric in y and z.
Consider thirdly the case z e A*, j/, z 6 A^. The definitions give (x, y o z) =
-x{yz) 6 A* = C, ([x,y\, z) = —{xy)z e A* = C. This proves the relation in all
cases.
Next, we notice that the definition of i is (for x e A) ia; = J] i'^ ® x'^ G
Q
W (QW where [x,y\ - T.^'a{^'a<y) for ^^l y eW. This says that ([x,y],z) =
Q
X;(xQ,z)(Xa,y) = [ix,zi^y). We have already seen that the left hand side is
a
symmetric in y, z so this gives
[x,yoz) = {[x,y], z) = (ix, y®z)w%w , (*)
which shows that the map o is invariant under A. It also shows that ix lies in
the space of symmetric tensors because [ix, y®z)=- [ix, zi2>y).
Finally we rewrite (*) as
[x,ot) = [ix,t)w»w, (tew®w),
so on substitution, we have (i, o is*)^ = [ix, id)w^w ~ 12(x, x')a- Since this is
true for all x we get o 23/ = 12i'. This finishes the proof of Theorem 12.4. []
Corollary 12.5. The action of ?77 on W is faithful.
Proof. We have LlEj) as a summand in W®VK. Since all the above maps
preserve the action of L[E7) and Ej is comiected, they also preserve the action
of Ej. If geEj fixes W, it fixes W^SfW, so fixes L[Er) and L[E-, x SU[2)) C
L{Es). But it also fixes the complementary summand L[Es)/L[E-! x SU[2)) S
Wi^X', so g fixes L{Ei). Thus g = l m Eg. Q
? as a group of maps of W.
Cartan showed that W carries an invariant quartic function — equivalently an
invariant symmetric map W®W®W(8)W—>C. (This equivalence is analogous
to the equivalence of quadratic forms and symmetric bilinear forms: 6 gives q
by q{x) = b[x,x) and q gives b by b[x,y) = coefficient of 2A/J in q{\x + ny).
Similarly for n-linear functions / — we get F{x) = f[x,... ,x) an n-tic function
and if F{x) is n-tic we recover / by /(xj,..., Xn) = coefficient of n! Ai... A„
in F[\iXi + ¦ ¦ • +A„x„). This process is called polarisation. In general it is better
to work with the sjonmetric multilinear /.)

Direct construction of E-, g]^
We define an invariant symmetric quadrilinear function on W by
/{¦w® xiSiy iSiz) = l(^{w ox, yoz) + {iuoy, xo z) + {wo z, X a y)Y
The next theorem says that we can recover {w o x, y o z) from /.
Theorem 12.6. {wox, yoz) = f[w®x®y®z)-\{w,y){x,z)~\{w,z){x,y).
The proof requires lemmas: the first helps us work out {wo x,y o z).
Lemma 12.7. If a,c e A^ h',d' e X^' then
(oo6*, cod') = iF*,o)(d*,c) + F*,c)(d*,a) - (bV,oc).
Proof. Use the standard operating procedure. f]
We can now evaluate / explicitly.
Lemma 12.8. If a,b,c,d e A^ a',6',c*,d* e A^* then
/(a®6®c®d) =abcdeX^{V)^C,
f{a®b®c®d') =0,
/(o®6®c'®d*) =\{c',a){d',b) + \{c',b){d',a)-{c'd',ab},
f{a®b'®c'®d') =0,
f{a' ® 6* ® c* ® d') = a'b'c'd' G A»(V) = C. Q
Proof of Theorem 12.6. This follows from Lemma 12.8. Q
Corollary 12.9. Let G be the identity component of the group of linear
maps g -.W —>-W which preserve ( , ) and Cartan's invariant quartic function.
Then G = Ej (and G aJso preserves the associated quadriJinear/unction).
Proof. G is a Lie group because it is a closed subgroup of a linear group.
We have a homomorphism Ej —>¦ G given by the action of Ej on W and it is
mono by Corollary 12.5. G acts on Hom{W,W) by g{9) = g6g-\
Claim. This action preserves the subspace A C Hcnn{W,W).
Proof. Invariance gives {gwogx, gyogz) = (woi, yoz) from Theorem 12.4.
Thus {\gwogx, gy], gz) = ([woi, y], z) = {g[wox, y], gz) for all gz and since
( , ) is non-singular, we have g[w o x, y] = [gw o gx, gy], so regarding these as
linear maps defined on gy we get g{w o x)p~' = gw o gx & Hcnn{W,W). But
elements of the form w o i generate A by Theorem 12.4. Hence gAg~^ = A. []
Clearly 9 \-^ gdg'^ preserves commutators, hence there is a map G —>/lut(A)
with kernel = {± 1}. (Proof, g : W ~* W is in the kernel if and only if g

82 Chapter 12
commutes with all 9 G A so g9g~^ = 9 =^ g = \ by Schur's Lemma because A
is irreducible, and since g preserves ( , ) we must have A = ± 1. Clearly the two
maps ± 1 : W —> W do preserve all structures and yield the identity element of
Aut{A).)
Now consider E-, -^ G —* (the identity component of Aut[A)). As the Killing
form on A is non-singular, by the Addendum to Theorem 12.1, this composite is
epi, and 1, i e 5Z7(8)/{± 1} form ker(G —> the identity component of Aut{A)).
We have as a consequence that Et —* G is epi (and mono by Corollary 12.5), so
E, = G. D
Real forms of Ey.
First we deal with the compact form which is what we obtain as a subgroup
of ^g. The invariant inner product we had on the real vector space L{Es) was
positive definite; it has to restrict to an invariant inner product on the real vector
space LlEj) and this must be a scalar multiple of the one we have already. The
one we have already was chosen negative definite on L{SU{8)), so the multiple
has to be negative. The complementary summand L{Ey)lL{SU{S)) gives a real
vector space U of dimension 70 whose complexification is the space A"* considered
above; on this space U the inner product of L{Ei) is real and positive definite,
therefore the inner product of ?'(?'7) is real and negative definite. The Killing
form has signature —133.
For the second form of Ey, we work through the construction of the last
chapter, taking V to be a real vector space of dimension 8, L to be L{SL{V))
and interpreting A* as a real vector space. We get a real Lie algebra L + X*.
The Killing form is still 36 times the inner product, and it has signature 7 on
L, 0 on A*, giving a total of 7. Under this construction we see that A' + A^* is
a representation of this version of Ej.
A third possibility is to go back to the subgroup Spin{12) x Sp{l)/Z2 C ?7
and consider the element —1 e Spin{12). It acts as 1 on L{Spin{12) x 5p(l))
but as —1 on the complex representation A"*"® A'. Let its —1 eigenspace on
the real space L{Ey) be U'. Then L{Ej) = L{Spin{12) x 5p(l)) + U' and
L{Spin[12) X Sp[l)) + ill' is a new real Lie algebra. Its Killing form is negative
definite on the first summand, of dimension 69, positive definite on the second,
of dimension 64, so the signature is —5.
A fourth possibility rests on the existence of ?^6- We have a map E^ x SU{3) —>
Eg. We can also construct a map S^ x SU{2) —> SU{3) mapping the circle into
diag{z'^,2~^,z''^) and SU{2) onto the last two coordinates as usual. So we get
?¦6 X S' X St/B) -* Eg. Since Ee x S' centralises SU{2), we have a map
^e X S' -+ Ej. We wish to know the restriction of the representation L{Es). We
have
L{SU{3)) ^ L(S' X SU{2)) + (r,^ + r^)\i ,
where tj is the identity representation of S'; A3 h-> 77^ + r}~^X\, A3 i-> r}'^ + r^Xl.

Direct construction of Ej 83
Thus letting M be the 27-dimensional representation of Eg, we get
L{Es) ^ LiEsxS' X SJ7B)) + (j73+r)-')Aj + M®(j7^+J7-'Aj) + M®(r)-^+ r)Aj).
Taking the part fixed under SU{2), we see that
?,(?'7) H-» L{E6 X S') + M ® 7)^ + M ® T)-^
The element i G S' acts as 1 on L{E6 x S') and as —1 on the complementary
summzind. Let the —1 eigenspace of i acting on the real space ^(EV) be U".
Then L{Ey) = L{EexS^)+U" and L(E6 x S')+it/" is a new Lie algebra over
R; its complexification is still C ® L{Ej) and the signature of the Killing form
is -79 + 54 = -25.

Chapter 13
Direct treatment of Eq
The object of this chapter is to do for Eg what Chapter 12 did for Ey. We use
three vector spaces Vi, V2, V3 of dimension 3 over C and their duals V', Vj*,
V3. Choose an isomorphism A'(Vi) = C, its dual A'(Vi*) = C and use these to
make the identifications A'(V^) = V^* and \''{V^} = Vj. For ii.u; G Vi, we have
an anti-symmetric bilinear product vweV^ and similarly for v' ,iu' eV'.
Lemma 13.1. We have
u'{v'w) = {u','w)v — {u',v)'w,
(vw)u* = (u*,d)w - {u','w)v,
u{v''w') = (w*,u)t;* — {v',u)iu*,
(i;'u;')u = (ii*,u)w* - {w',u)v'.
Proof. For (R')* = B? these are well-known properties of the vector
product and they are proved using the standard operating procedure. Q
Construction of Ee and its 27-dimensioncil representation.
Everything we do will be invariant under SL{Vi) which we identify with
SL{V^). Put
Li = L{SLm)) = L{SL{V^)),
L =^Li + L2 + Li = L(SLC,CK) = L{SU{3f).
As a candidate for L{E6), set
A = L -I- Vi ® V2 ® V3 -I- v; ® v; ® ^3* = L -I- ViViVi + v^'V^'V^',
dropping the ®, and as candidates for the 27-dimensional representations,
W ^V.V.' + V.V^' + V^V,',
85

86 Chapter 13
W' = ViV,' + V^V{+V,V^'.
(Compare Theorem 11.4.) We may also identify V^\A' with Homc{Vj,Vi) and
hence W and W with linear maps of Vj + V^ + V3. Thus W is the space of
/0A0\ /00?)\
matrices 0 0 B j and VF* is the space of matrices ? 0 where the
\C 0 0 ) \ 0 F 0 J
blocks are 3x3. The duality of W and W is displayed by the map W®W' -* C
given by w® w* 1—> Tr{w'w) where w, w' are viewed as the matrices above, or
by UfUj®VjV' i-> {Uj,Vj){Vi,Ui). The action of A on W^ and W is given as
follows: L^Li + Li+Ls acts as L{SL{Vl}) ^ L{SLiV^)) acts on K and V^*;
U1U2U3 e VjViVs acts on ViVj e ViV/ to give (t;*, «y)Ui ®u,Ui ? ViV;' (A; ^ i,j).
Similarly ujujwj 6 ^i*V'2*^3' acts on ViVJ G VJV/ to give {ul,Vi)ujvj(Sul e V^l^*
Lemma 13.2. The pairing W^' ® W —* C is invariant under this action.
Proof. Direct from the anti-symmetry of the triple bracket A^(Vi) = C. []
Theorem 13.3. These maps embed A as a Lie algebra of linear maps W —*
W (and W' —> W). This action coincides, up to isomorphism, with the action
of L(Es) on one of its 27-dimensional representations.
Proof. Embedding; (As for Ej, see Theorem 12.1.) A is the sum of
inequivalent irreducible representations Li, L2, L3, 14^2^3 and V^'V^Vj of
YlSL(Vi) and the map into HoTn{W,W) is non-zero on each,
t
Next we show that the image of A —* Hom{W, W) is closed under
commutators.
It is clear that this is true if a or 6 G L because all our constructions are
invariant under L. We see that the bracket in A of an element in L and an
element in V1V2V3 is given by the usual action of L on VjViVa. Suppose now
iit2ti,UiU2U3 e V1V2V3. We wish to define [x,y] where i = tit2^3,J/ = WiM2W3- If
L is to act as a Lie algebra on W, we must have [x,jf]v = x[yv) — y{xv), for
V eW. The definitions give
1*1*2*3, [U1W2U3, ViV'j]] = (ll*, Wj)[tit2i3, UklSlUiVi] = (ll', Uj){tiU,Vi)tj (^1 t^Uk.
Interchanging the t's and u's gives
[U1U2U3, [*1*2*3, ViV']] = («•, tj)[UitiVi)Uj®Uktk = [v], tj}{tiUiVi)UjlS)tkUli.
The obvious element to try in V^'V^V^ is (iiUi)(*2i'2)(t3U3) and the action
of this on Dji)* is:
[(*l«l)(*2«2)(t3'^3), ViV'^] = {tiUiVi}{tjUj)vj IS) tkUk
= {ii'^iVi){v'j, tj)uj ® tkUit - {tiUiVi){v^, Uj)tj ® ttUk,

Direct treatment of Eg 87
where we have used Lemma 13.1. We see that we should take
[hhta, ¦UiU2«3l = -{tlUi){t2U2}{t3U3}.
The case when f;, Ut are replaced by t', uj is exactly similar.
Finally take tjtjtj G V-^V^V^, U1U2U3 G ViV2V3. From these elements we get
an obvious element in L and its component in Li is {tj, Uj){tl, Ui;)n{ui (gi t*),
j ?" ^t J ?" h k ^ i, where 7r : Vi (g) Vj' —> Li is the projection defined by
7r(i) =x-\{Trx)I.
The commutators in Hom{W, W) satisfy the Jacobi identity, therefore so do
the ones in A and we conclude that A is a Lie algebra.
To identify A with L{Es) and W with the 27-dimensional representation,
we choose Hermitian forms on V1V2V3, obtain groups SU{Vi) and identify these
with three copies of SU[Z) in SU[2>f -^ Eg.
Now choose embeddings V1V2V3 — LiEe), ViV^', V2V'3*, V3V1' -^ W (the
27-dimensional representation) compatible with the action of SU{2)'^. We still
have the freedom to multiply these embeddings by Q,/3,7,<5 G C respectively.
The action of V1V2V3 (in L{Ee)) on these parts of W must coincide with the
above constructed action on ViV' up to scalars (which are non-zero). We can
vary these scalars by aP/6, a^y/P, 06/7 and so choose q to get the product of
the three scalars to be 1. Choose p, 7, 5 to get all of them 1. We thus identify
W and W so as to match up the action of L and Vj V2V3. Since A is generated
by L and ViV^Vs under brackets, it follows that A C Hom{W,W) is contained
in Im{L{Ee)) C Hom{W',W') = Hom{W,W} and Im{L{Es)) is closed under
brackets. But dim A = 78 and dim(/m(L(?'6))) < 78. Hence A = lTn{L{E6)).
D
Next we look for whatever other structure there is around. We have a
symmetric bihnear form on A: on each Li, it is to take matrices X, Y to Tr{XY).
For elements ujujuj G V^VjVj, i'ii'2i'3 G V1V2V3, we set
(ujujuj, 111112113) = - (u;, Vi){u'2, V2){U3, V3).
The remaining summands are to be orthogonal.
Lemma 13.4. This symmetric bilinear form ( , ) on A is invariant and the
Killing form is 24( , ).
Proof. Straightforward. Q
In the following lemma we use this inner product to transpose structures,
compare Theorem 12.4.

88 Chapter 13
Lemma 13.5. There is a unique map W®W'^^A such that (a, wow') =
([a, w], w'). It is invariant under A and we have a commutative diagram:
Hom{W,W) '^W®W'
I
o
A
Proof. Since the inner product in A is non-singular, it is clear that this map
exists and is unique. It is invariant since ail the ingredients of the structure are.
For later use we give the map explicitly as follows :
on Vi14' ® 14Vj*, we have Ujuj o V2vl = (wj, J/2)t(wi ® Uj') — (i;,*, UiOr(ii2 ® uj),
where 7r(x) = x — ^Tr{x)J and u^ ® v' is to be considered as a matrix,
on ViV^ i®V3V2,-wehave UiU^ovzv^ =-UiiSlu^vji^vj eViV2V3, and
on V1I4*® I/1I4*, we have u^ul QViv^ =—UiVi<SU2®vj eViV^'Vj. Q
Corollary, The action of ?5 on either 0/its 27-dimensiona] representations
is faithfui.
Proof. (As for Ey, Corollary 12.5.) Suppose g e Es C Eg fixes W,
the 27-dimensional representation. Then by duality it fixes W so also fixes
W® W. But L(?'e) is a summand of W® W*, therefore g fixes L{Ee) and
L{Es X St/C)) C L[Ei); it also fixes the complementary summand L{Es)/L{E6X ¦
SU{3)) S ty ® A' -I- M^* ® A^ so it fixes L{Es). Thus ^ = 1 in ?;8. Q
Next we define two bilinear maps W®W^^W', W'®W'^^W as follows:
on VIV; ® ViV;, u,u' x ViV' = -(u'l';) ® (u,i;,) e VjV;,
on ViV/ ® Vj.Vj* (i,j, k distinct), UiU* X VjV^ = (w^^ Uj)u,- ® uj ? V'iV^*, and
on V,V; ® KV^', v,vl X «iw; = (u',i;^)u, ® 1;^ e v^v;.
Hence X is symmetric. We can also express this as follows.
If j4 is a 3 X 3 matrix, then Adj A is a quartic function of A so we may
polarise: Adj{iJ.A + vB) = fj^AdjA + 2nv[A x B) + i/^AdjB. The product on
matrices is defined by
f 0 A Q \ f 0 D 0 \
\qob\x\ooe\
\C 0 0 J \F 0 0 )
( 0 0 AE + DB-Cx F\
= \ BF + EC - Ax D 0 0
V 0 CD+FA-BxE 0 /

Direct treatment of Es 89
Lemma 13.6. These products are invariajit under L{E^), therefore also
under Es. g
The following lemma says we can recover the inner product W* ® W —> C
from X.
> W is 10(u;',u;) and similarly
D
We proceed to introduce trilinear functions / : W ® W (^ W —^ C and
/ ; W ® W ® W -> C by /(x ® y ® z) = (x X y, z).
Lemma 13.8. The functions f are symmetric in ai/ three variables.
Proof. On ViV- ® Vi^/ ® Vy^', we have /(UjU* ® v^v'^ ® w^wj) =
— {u^v^Wi){UjV'Wj), where both factors are anti-symmetric. Otherwise a sum-
mand on which / is non-zero is, up to rearrangement of factors, Vi'V^Vj'Vj V^'V^,
and f{vlVii^v'2Vj®v'3V3) = {vl,Vi){v^,Vj){v'3,V3). []
Since the trilinear functions / are symmetric, we lose no information by
considering the associated cubic function c(x) = /(x® x® x). We give it in matrix
notation.
Lemma 13.9. / 0 A 0 \
c 0 0 B =6(Tr(A5C)-detA-detS-detC),
V C 0 0 /
/ 0 0 D \
c ? 0 0 =6(Tr(F?;D)-detD-det?;-det F). ?
V 0 F 0 /
Everything else can be reconstructed from the two cubic functions and the
pairing, but as polarisation is inconvenient, it is better to retain formulae for the
cross products. The next lemma is the key in recovering everything from the cross
product.
Lemma 13.10. For x,z e W, y',z' e W' we have
[xoy',z] = {y',z)x + ^{y',x)z-{xx z) xy', and
[xoy*, z'] = -(z*,x)y' - ^{y\x)z' +xx {y' x z').
Proof. It is only necessary to prove one formula as the other follows by
transposing the first using the fact that {z,z') is invariant under the action of
X oy'. The verification of either formula is tedious. To check the coefficients on

90 Chapter 13
the right-hand side, note that [xo j/*, z] is a linear function of 2 and so the trace
of this linear function should be 0. On the right-hand side, the trace of z i—> z
is 27 and that of z >-> {x X z) X y' is 10(j/', x), by Lemma 13.7. []
At this point, starting with W, W* we have a possible alternative way to
construct a Lie algebra of maps W —> W. Start with the products x : W® W —>
W', X : VV*® W* —> W, and recover the pairing W*® W —» C ?is before. So we
know the right-hand sides of Lemma 13.10. Unfortunately, with this construction,
to check that we have the required properties, we need some more properties of
the cross product. From the cross product we can construct several maps from
W* —» W with various symmetries, but up to a scalar, we have only one map
symmetric in all four variables.
Lemma 13.11. For w,x,y,z ? W, we have a symmetric map W* —> W
given byx®j/®z®wi-->
/(x® y® z)u; -l-/(w® y® z)x + /(w® x® 2)j/ -l-/(w® x® y)z
= {w X x) X {y X z) + {w X y) X {xx z) + {wx z) X {xx y).
Similarly for four variaWes in W.
Proof. W ® W -^ W* is invariant under the action of [x o j/*, ], so
[xoy',w X z]= [x o y', w] X z + w X [x o y', z],
[x o y', w] X z + w X [x o y', z] — [x o y', w X z] — 0.
On substituting values from the formulae of Lemma 13.10, the left-hand side
of this equation gives
{y',w)x X z + ^{y',x)w x z — {{x x w) x y') x z
+ {y',z) w X X + ^{y',x)w x z - w x {{x x w) x y') ¦
+ {w X z, x)y' + ^{y',x)w x z - {x x {y'{vj x z))) =0 6 W*.
Take the inner product with y ?W to get
(y',z)/(w®x® y) -l-(y',x)/(uj® y®z)
+ {y',w)f{x®y®z) + (y,y')/(w®x®2)
- /(y ® w ® (x X z) X y*) - /(y ® z ® (x X w) x y*)
- /(x ® y ® (u; X z) x y*) = 0,
which leads to the result when we remove y', since the above is true for all y'. Q

Direct treatment of Eg gi
Lemma 13.12. For all x,y,w ? W,
w o (x X J/) + X o (y X w) + y o (w X x) = 0,
and for w,x,y ? W,
{w X x) o y + (x X y) o w + {y X w) o X = Q.
Proof. Since A —> Hom(W,W) is mono, it is sufficient to prove that the
first expression gives 0 when it acts on any z ? W. When we substitute from
Lemma 13.10 to find the value of [s o t*, z], we get an answer which vanishes by
Lemma 13.11. SimilEirly for the second expression using the action on W*. h
Ee as a group of maps.
Finally, we wish to show that the symmetry group G of the structure we have
found is exactly E^. Our structure consists of two vector spaces W, W* and two
products W ®W ^UW and W ®W' ^^W. Then an element g will be a
pair of linear maps 71 : W —» W, 72 : W* —> W such that for all x, y ? W, or
x,j/ e W*, g{x X y) = gx X gy where the action of g on W is by 71 and on
W by 72.
Lemmia 13.13. The symmetry group G preserving these products is Es.
Proof. It is clear that an element g must preserve the pairing W*®W —>• C.
In fact the trace of the composite
wx w'x
w—>w—>w
is equal to the trace of the composite
(su))x (s"'')x
W— >W'~—-^W,
that is 10(w',u;) = lQ{gw',gw), by Lemma 13.7.
Next, the subspace A C Hom{W,W) is closed under a 1—> gag~^. To check
this, from Lemma 13.10 we have
[xoy', z] = {y',z)x+^{y',x)z-{xx z) x y'
and this gives
g[xoy', z] = [gy\gz)gx + \{gy', gx)gz - [gx x gz) x gy' = \gxogy\gz].
Thus
g[xoy\g-^z] = \gxogy', z],

92 Chapter 13
for all x,z&W, y'&W. So, for a = xoy' the map a>—^gag~^ lies in A and
as such a generate A, A is closed under a i—> gag'^. The endomorphism a >-*
gag'^ of A preserves commutators, so we get a homomorphism G —> Aut{A).
Its kernel consists of the elements g such that the action on W commutes with
the action of all a ? A. Since W is an irreducible representation of A, g must be
multiplication by a complex scalar a and since g preserves the function f[x®
y®z), we must have a'' = 1 so a = l,ii;, w^. Clearly these maps do preserve all
structure and 3neld the identity element of Aut[A). Consider the composite E^ —>
G —* (the identity component of Aut{A)). The Killing form on L{E6) is non-
singular by Lenama 13.4, so the map Ee —>¦ (the identity component of Aut{A))
is epi. Moreover, there are enough elements in the kernel of this map to account for
1, w, vp- in ker(G —> Aut(A)). That is to say, we have an inclusion SpinilQ) x
[/A)/Z4 —> Es and the elements of order 3 in C(l) do what is required. Thus
the map Ee —* G is epi. It is mono by the corollary of Lemma 13.5. []

Chapter 14
Direct treatment of F4, I
In the previous two Chapters, we studied E^ and Ej by way of their lowest
dimensional non-trivial representations. Here we do the same for F4. We relate
the Lie brackets in L{Es) to the brackets in L{Fi),L{G2) and structures on the
26-dimensional representation U and the 7-dimensional representation V of F4
and Gj respectively. We also construct an algebra structure on U. In Chapter 16
we will construct U, together with all accompanying structures, independently
of this Chapter and show that F4 is indeed the group of automorphisms of U
preserving these structures.
F4 contains Spin{9) as a subgroup of maximal rank:
F4 D Spin{9) D Spin{8) D T.
We use the usual notation of X\, x^, X3, 2:4 for weights. From Chapter 9, the
roots of f 4 are
long roots : ±Xi±Xj, ' < J, 24 roots of Spin(8),
short roots : ±Xi 8,
|(±Xi ±X2 ±X3 ±14) 16.
Before going on it will be handy to introduce a pcirticular set of subgroups of
Lemma 14.1. The group F^ has a subgroup of local type 5GC) x 5GC)
so that the roots corresponding to the first /actor S[/C) are Jong roots of Fi and
the roots corresponding to the second factor are short roots.
(Note that these SC/C) are not in Es-)
Proof. Consider 5GC) -^ 5pinF) -^ Spin{9) — F4. This 5GC) has roots
±(xi — Xj), 1 < i < j < 3, and its L{T) is determined by Xi + Xj + X3 = 0,
93

94 Chapter 14
X4 = 0. Now consider its centraliser in F4; its L[T) is defined by Xi = Xj = X3
and its roots are
±|(Xi +X2 +X3+X4)
±|(Xi +X2 +X3 -X4)
±X4
weights for A"*"
A-
X\
This centraliser is of type S[/C) which inherits its roots from the short roots of
F4. Thus we get a map 5GC) X 5[/C) —> F4 as required. Q
Theorem 14.2. The Weyl group of f\ has order \W{Fi)\ = 2^ • 3^
W{Spin{8)) is a normal subgroup of W{Fi) with W{Fi)/W{Spin{8)) = E3
ajid this extension is split. The quotient E3 preserves Spin{8) — indeed it is
Out{Spin(8)). Thus all automorphisms of Spin{8) become inner in F^.
Proof. Consider x ? N{T) C F4. Then y i-> xyx'^ gives an inner
automorphism preserving T. Clearly it carries long roots to long and short roots to
short, so preserves
L{Spin{8)) = L{T) + {eg\9 long),
where e^ is an eigenvector corresponding to 6. Therefore it preserves Spin{8).
In particular, it preserves W[Spin[8)) which shows that W{Spin{8)) is normal
in W{Fi).
To continue, we have a map
N{T) -* Aut{Spin{8))/Inn{Spin{8)) = Out{Spin{8)) = S3.
Since clearly T maps to 1, we get a map
W{F^) = N{T)/T -^ Out{Spin{8)) = S3.
The kernel is W{Spin{8)). In fact, it is clear that an element of W{Spin{8))
maps into Inn{Spin{8)). Conversely, suppose given an inner automorphism y i—>
xj/x~' of F4 which restricts to an inner automorphism y 1—> zyz~^ of Spin{8).
Then its efltect on T cam be given by y 1—> 2j/z~' in Spin(8), i.e. we had an
element of W{Spin(8)). Hence we get a monomorphism W(F4)/W(Spin{8)) —>
E3. This map is split epi. For, consider the subgroups of type 5GC) X 5GC)
of Lemma 14.1. We get W{SU{3) x 5GC)) C ^^(^4) and in particular ^(^4)
contains a S3 subgroup given by the Weyl group of the second factor, the one
which inherits its roots from the short roots of F4. This S3 acts by permuting
the root pairs
±|(Xi +X2 -f X3+X4), ±|(xi+X2 + X3 - X4), ±X4.

Direct treatment of F4, I 95
Therefore, this S3 permutes the representations A"*", A~, A' of Spin[8) in the
same way. n
To continue, let U denote the 26-climensional representation of F4 and recall
that
(i) U is real;
(ii) U\Spin{9) is l + A'+A;
(iii) as a representation of F4 x Gj we have
L{Es) = L{F,) + L{G2)+U®V,
where V is the 7-dimensional representation of Gj arising from Gj C Spin{7).
Thus U\Spin{8) is 2 + A'+ A++ A" and in particular, its weights are the
short roots of F4 together with zero (twice).
Lemma 14.3. The representation U of F4 is irreducible.
Proof. The non-zero weights form a single orbit under W{F4,) so the only
possible decomposition for U is U = 1 + U', where [/' i-> A' + A on Spin{9).
Restrict to the subgroups of local type 5[/C) X SU{3) of Lemma 14.1. Then, by
consideration of weights, we see that [/ 1—> 1 ® adj + \^®X^ +X^® A'^, where the
parts are of dimension 8, 9, 9 respectively. The two zero weights are in 1 ® adj
which, nevertheless, is an irreducible representation of SU{3) X S[/C). We see
that U must be irreducible. Q
Lemimia 14.4. (i) As a representation of F4, we have L{Es) = L{Fi^)-\-U, so
U can be constructed as a representation of F4 on L[Es)/L{Fi) (giving another
proof that U is real).
(ii) Both the 27-dimensiona] representations 0/Eg restrict over F4 to \+U.
(iii) On restriction from Eg to ^4, 24 of the 27 weights 0/each 27-ciiniensionai
representation restrict to the 24 short roots of F4. The remaining 3 restrict to
zero and these 3 add to zero before restriction.
Proof. We have
L{Ea) = L{Fi) + L(G2) + U®V over F4 x G2 ;
L{Eg) = L{Fi) + L{SU{3)) + A' -t- A^ + [7 ® (A> + A^ + 1) over F4 x St/C);
L{Es) = LiEe) + LE[/C)) +W®X' +W'®X'' over Eg x SU{2),
where W and W Eire the 27-dimensional representations of Es , see Chapter 13.
Thus taking the part fixed under S[/C), we get L{Es) = ^(^^4) +U, and taking
the coefficients of A' and A^ we get W,W ^-^l + U, which proves (i), (ii). To
finish the proof of (iii), we need only check that the three roots add to zero before

96 Chapter 14
restriction. With explicit calculation and our usual coordinates for f\, we see
that the three weights which restrict to 0 on F4 are X5 +a;6i ~^i +^6, —Sle for
one representation and minus these for the other. (Note that xe, 27, Xg all give
the same function on T C Be-) Q
Structure maps on U.
We are now ready to see what sort of structure maps the representation U
carries.
Theorem 14.5. (i) TTie representation U carries a symmetric biiinear map
b : U ® U —^ C invariemt under F4 and non-singuJar. (This follows from the
reality of [/.)
(ii) The representation U carries a symmetric biiinear map U ® U —>• U
which is invari«int under F^ and epi.
(iii) Moreover, the function t/^ —> C, t(u',u",u"') = 6(u'xu", u'") is triJinear
and symmetric in all three variables.
Proof 1. This uses our knowledge of Eb-
Let W be the 27-dimensional representation of Eg. We know that there is
a non-zero / : W ® W ®W —> C, symmetric and invariant under Eg- By
restricting to F4 we get a function / : A + C/) ® A + [/) ® A + [/) -> C,
which is symmetric, invariant under F4 and non-zero. All we need to know
now is that /:C/®C/®f/—>C is non-zero, for if it is and we define X by
6(u' X u", u'") = f{u' ® u" ® u'"), we get a product X : U ®U —> U invariant
under F4, non-zero and epi since U is irreducible. To show that / 7^ 0, recall
from Chapter 13 that W = Vj Vj* + ViVs' + V3V1'. Let the weights of
Vi be Xi, Ea;; =0,
V2 be yi, Eyi = 0,
V3 be z., Ez. = 0.
The weights of W are x; — j/j, j/j — Zk, z^ — Xi.
The triples of weights adding to zero are
(i) X, - j/j, Vj - Zk, Zk-Xi]
(ii) Xi-y,, Xk-yt, Xm-yn, where {i,fc,m} = {1,2,3}, {j,?,n} = {1, 2,3}.
Whichever triple restricts to 0 on T{Fi), we can find another such {9,(p,ip}
disjoint from it. Then the corresponding eigenvectors e^, e , e^ will lie in the
summand U of W = l + U and we have /(e^ ® e^ ® e,^) ^ 0.
Proof 2. This requires some preparation and will use the fact that f 4 =
CEaiGi), the centraliser of G2 in Eg, see Chapter 8.

Direct treatment of Fi, I 97
Lemma 14.6. As representations of Gi, we have \''{V)^L{G2) + V, where
G2 C Spin{7) acts on R^ giving a 7-ciimensionai representation V.
Proof. Restrict to 5GC). Then
V H-* Ai + A^ + 1,
L(G2)>- L{SU{2,)) + \'+X\
X''{V) ^ A^ + A'+O + A'A^ + A'+A^
= L{SU{3)) + 2A' + 2A2 + 1 = L{Gi) + V.
Thus A^(V) and L(G2) +V are isomorphic over 5GC), and as 5GC) is of
maximal rank in Gi, they are isomorphic over Gj- Q
Note also that we have a product V ® V —> L{Gi) given by (a, u o w) =
([a,!)], w). One checks that o is anti-symmetric.
Corollary 14.7. There is a skew symmetric product V®V—yV iiwa,naiit
under G2 and it is unique up to a scalar factor. (This is obvious if we regard V
as the space of pure imaginary Cayley numbers.)
Proof. The representation V is irreducible; in fact, from its weights, it could
only split as 1 + V. We know that G2 is transitive on S* C V. Also L(G2) is
irreducible and disjoint from V so that we have a G2-map A^(V) —> V, using
Lemma 14.6. Q
Lemma 14.8. Any sJcew-symmetric map jj, : U ®U —* U mvariant under
Fi is zero.
Proof. Restrict to 5pin(9). Then G i->-1 + Aj + A, so
a2(G) h^ A^ + A2(A) + A> + a + A> ® a = A^ + (A' + A^) + a' + a + a' ® a.
(See Theorem 4.6.) Thus U hsis a suramand 1 and A^(G) does not, so A^(G)
does not have a U suramand. []
Now consider L{Es) = L(F4) -I- L(G2) +U ®V. We put an inner product
on L{Es), non-singular, symmetric, bilinear and invariant under Eg. The three
summands are to be orthogonal. Moreover, since U and V are irreducible, there
is, up to a scalar, only one way to put a bilinear form on G ® V: we must have,
for s,t e U, v,w e V, (s ® u, t ® w) = b{s,t){v,w)y where ( , )v is some
inner product on V, invariant under G2 and 6( , ) is a bilineM form on G,
invariant under F4. We know the Lie brackets on ^(^4) and L(G2) and wish
to determine other brackets. Well, ^(^4) acts on G ® V via its action on U
and L(G2) acts on U ®V via its action on V. There remains the Lie bracket
(G®V')®(G®V^) —> L(?8). y4 priori, this has three components: one into L[Ft),

98 Chapter 14
one into LiGi) and one into U ®V. For present purposes, we are interested in
the third.
Lemma 14.9. The component of the Lie bracket (U (8) V)^ —> [/ ® V is
[s ® V, t ® vj] = {s X t) ® {v X w)
where x : V®V —>V is as in CoroJJary 14.7, and X : U ®U —^ U is symmetric,
invariant under F^ and epi.
Proof. The Lie bracket is a skew-symmetric map
for some fj -.U^U —^U invariant under F^, symmetric by Lemma 14.8, and
5i : V ® V —> V invariant under G2 and skew-symmetric. By Corollary 14.7,
each Qi = XiQ for some scalar Aj, so '^fi®gi = f®g where / = HAj/j. It
remains to prove that / 7^ 0. Recall that if 9,ip, 6 + ip are roots of Eg and e^,
e^, fij^^ the corresponding eigenvectors in L[Ei) over C, then [69,650] = Ae^^^
for some A ^ 0. We want to choose 6, (p so that the restrictions to F4 and G2
Eire what we require. In fact, taking
9 — \[x\ -I- X2 + X3 -I- X4 + X5 -I- X6 - 2:7 - 3^8),
^ = \{~^\ - X2 — X3 + X4 - X5 - X6 + 2:7 - a^s),
we have 9 -I- (,0 = X4 — xg, and e^, e^, e^^^ lie in [/ ® V over C. Hence [ , ]
is non-zero. This shows that X : U ®U —^ U is non-zero. Since it is invariant
under F4 and U is irreducible, it must be epi. Q
It remains to prove the third clause of Theorem 14.5, but we will prove a bit
more. Let us use s, t, u for variables in U and v, w, x for variables in V.
Lemma 14.10. (i) b{s x t, u) =^ 6(s, t X u) and b is symmetric in all three
variables.
(ii) (i; X w, x)v = {v, w X x)v Bjid is an aJtemating /unction of its three
variables.
Proof. The inner product in L{Eg) is invariant, so
([s®ii, t® w], u® x) = (s® u, [t® w, u® x]), that is
((s X t)® (i; X w), u® x) = (s® u, (t X u)® (w X x)).
Hence
6(s X t, u){v X w, x)v = b{s, t X u){v, w x x)v. (*)
Thus the functions s® t® u 1—> 6(s x t, u), s® t® u 1—> 6(s, t X u) are non-zero.
Similarly, v(B>vJ®xt-*{vXw, x)v, v®w®xi-->-{v,wx x)v are non-zero.

Direct treatment of Fi, I 99
If either pair of functions were linearly independent, the other pair -would
be zero, contradicting (*). So in each C?ise, the two linear functions are scalar
multiples of each other and the following six functions are non-zero scalar multiples
of each other:
b{s X t, u), b(t X u, s), 6(u X s, t), b{t X s, u), b{u X t, s), b{s x u, t).
We get a 1-dimensional representation of E3, which is therefore trivial or the
alternating representation and since 6(s X t, u) = b[t x s, u), it must be trivial.
The same argument gives that the following six functions are non-zero scalar
multiples of each other:
{v X w, x)v, {w x: X, v)y, (x X V, w)v, {w X V, x)v, (x X w, v)v, {v X X, w)v,
and since {v x w, x)v = —{w x v, x)v, the representation of S3 is the alternating
representation. []
This finishes the second proof of Theorem 14.5. Q
One can go much further with the properties of x, 6 and ( , ) using for
example the Jacobi identity in L{Es). Thus one can define
o:U®U-*L{Fi), by (a, sot) =6([a,s],t),
o: V® V-* L(G2), by {v', v ow) = {[v',v], w),
both of which are anti-symmetric. The remaining two components of the Lie
bracket (U ® V)^ —> L{Es) can be given in terms of these products (we gave the
component into U ®V in Lemma 14.9). They are
{U'SiVy -^ L[Fi), {s®v)®[t®w)^ {sot){v,w),
-> L(G2), {s®v)®{tiSw)i-^b{s,t)vow.
The fact that these components can be given in this way follows from the invari-
ance of the inner product in L{Eg). Furthermore, we can describe the action of
sot on U and of now on V: we must have
[so t,u] = a{b{t,u)s — b{u,s)t) + P{{t x u) x s - (u x s) x t), (**)
where q, /3 are scalars determined by various choices not made explicit.
Considered as a function of u, [sot, u] is a derivation with respect to x, that
is,
[s o t, u X u'] = [s o t, u] X u' + u X [s o t, u'],
so the right hand side of (**) is a derivation and we have a non-trivial algebraic
relation between b and X. Similarly,
[i; OH), x] = 'y{{w,x)v - {x,v)w) + S{{v X w) X x + {w x x) X v + {x X v) X w),

100 Chapter 14
where 7 and S are scalars, inexpiicitiy determined by various choices. As before,
[v o w, x] is a derivation with respect to x. We deduce from (**) yet another
proof that x is not zero. (If x were zero, we would have q ^ 0 so that the
linear functions of u in the a term would give us all the linear maps f/ —> i/ in
L{SO{U)) of dimension 26 • 25/2 = 325, and they cannot all come from ^(^4).)
The algebra structure on U.
We now seek to show that the 26-dimensional representation U is isomorphic
to an explicitly constructed algebra. (An algebra, over a field, is a vector space,
A, together with a bilinear map Ax A —> A.) The difficulty is with the 2-
dimensional part of U on which Spin(8) acts trivially. Clearly NF^{Spin{8))
acts on this 2-dimensional subspace and hence NF^{Spin(8))/Spin(8) acts on it,
so E3 = Out{Spin[S)) acts on it.
Proposition 14.11. This 2-diniensiona] part is the (reducedj permutation
representation of E3.
Proof. Let S[/C) X 5t/C) -> F4 be as in Lemma 14.1. Then, as in the
proof of Lemma 14.3, we have [/ !-> A' ® A' + A^ ® A^ + 1 ® LEf/C)) over
SU{3) X 5[/C). The 2-dimensional subspace is L{T) for the second 5GC)
(where the zero roots are). Our S3 is the Weyl group W{SU{3)) of the second
factor (see the proof of Theorem 14.2). The action of W(St/C)) on its L{T) is
the reduced permutation representation. []
We have E, = ^i^F^i^)) c ^(^^) = S,
we have h, ^^^p^^^g^) C ^(g^^^^g)) ^3-
Corollary 14.12. When the above Ej acts on our 2-<ijmensionai
representation, the +1 eigenspace is the summand 1 in C/i—>l + Aj + A over Spin{9)
and the —1 eigenspace is the summand 1 in AJ 1—> 1 -I- A| over Spin{8). Q
The next task is to take the structure on U and see how it looks if we break
it up according to the action of Spin(8). First, a lemma.
Lemma 14.13. Spin{8) is transitive on pairs [x, y) of vectors x E. S^ C A+,
yeS' CA-.
Proof. We assume that we have a positive definite inner product on A*,
invariant under Spin(8). Certainly Spin{8) is transitive on points 2 e S^ C Ag.
If we choose 2 = @,... ,C, 1) the subgroup fixing z is Spin{7) and A+ restricts
to A; Spin[7) is transitive on points x ? S^ C A by construction, so Spin{8) is
transitive on pairs {x, z), x ? S'' C A"*", z 6 S^ C A'. By triality, it is transitive
on pairs {x,y). Q

Direct treatment of F4, 7 101
Recall that we have a triality map / : A"*" ® A~ ® A' —> R, invariant under
Spin{S) and -1 < /(x ® y ® 2) < 1 when ||x|| = ||y|| = ||z|| = 1; these
bounds are attained. We thus find a product X : A"*" ® A" —¦A' such that
(x X y, z) = /(x ® y ® z).
Lemma 14.14. The above product satisfies ||x x y\\ = \\x\\ \\y\\.
Proof. First, let x, y, z attain the bounds, that is ||x|| = ||y|| = \\z\\ = 1
and /{x ® y ® z) = (x X y, z) = 1. But -1 < (x X y, z') < 1 for all z' with
||z'|| = 1. Thus we must have z = x x y, so for these x, y with ||x|| = 1 = ||y||,
we have ||x X y|| = 1. By Lemma 14.13, Spin{8) is transitive on pairs (x,y) of
vectors of unit length in A+, A". So we have ||x X y\\ = 1, if ||x|| = 1 = ||y||.
The general case follows by bilinearity. []
Corollary 14.15. R* admits the structure of a normed algebra with 1.
Proof. Choose base points xo ? 5^ C A"*", yo ? S^ C A", and identify
A" with A'
A+ with A'
= R»
by y !-> Xo X y.
by X 1—> X X yo.
With these identifications, x gives a normed product on R* for which oco X yo
is a unit. []
For the moment we put this Corollary aside because we refuse to break
symmetry by choosing base points xo, yo in this way. We move to representations of
Spin{9).
Lemma 14.16. (i) There is a triJinear map A®A®Aj—»RorC which
is invariant under Spin[9) and non-zero. It is unique up to a non-zero scaJar and
symmetric in the first two variabies.
(ii) Suppose given decompositions A <-^ A''"-|-A", Aj <^ Ag-l-1 over Spin{8)
preserving inner products. Then, up to a scalar /actor, the trih'near map becomes
(a++a-)®F++6")®(z-l-C) 1-^/(a+®6"®z)+/F+®a"®z)±C((a"^,''"^)-(a",6")),
where the sign is fixed for any set of choices, but can be changed by changing the
sign of any of A+ ->A, A" -> A, Aj ->¦ AJ, 1 -> AJ or /.
Proof, (i) By transposing, it is equally well to consider A® A —> Aj. But
we have cr^(A) = A' -I- A' + 1 and A^(A) = A^ + A^, by Theorem 4.6, so up to a
scalar multiple, there is just one such linear map and it is symmetric.
(ii) Because the map is invariant under Spin(8), the map in part (i), regarded
as a map A ® A —> Aj, must decompose as
(a+ + a-)®F+-t-6-)H^ (a(a+ x b') + p{b+ x a-),7(a+, 6+) + <5(a-, b')],

102 Chapter 14
where a, /3, 7, S are scalars and X : A"*" ® A" —> Ag is as in Lemma 14.14. By
symmetry a = p.
Consider the effect of an element g ? 5pin(9), covering fg J, where detA =
— 1. This element g acts on A and interchanges A+, A" preserving their inner
products. Suppose it carries a"*" of length 1 to a~ of length 1. Then we have
g®g
(o++0)®(a+ + 0)—@,7)
u
@,-7)
@-t-a-)®@ + a-) —»@,6) .
Thus 7 = —E and we are down to (aa* X b~+ab'*'x a~, ¦y(a'^,b'^)— j{a',b')j.
Now consider the 1-parameter subgroup joining 1 to such an element m
Spin{9), say gg = cos& + (e8e9)sin^. Acting on (a+,0) ? A we get, in A,
a path of elements 59(a"*",0) = (cos^)a"'" + (sin^)a', a~ 6 A", ||o-|| = 1. Under
our map A ® A —> Ag we have
((cos9)a+ + (sin(9)a") ® ((cos(?)a++ (sin&)o")
!->• f{2acos(9sinS)a"'" x a", 7(cos^ 9 - sin^(9)),
which must have constant length, because it is obtained by ge ? Spin(9) acting
on a fixed vector. But ||a+ X a~l| = 1 so that the square of the length of the
right hand side is a^ sin^ 29 +7^ cos^ 29. But this must be 7^. Thus a^ =7^, so
7 = ± a. , ?
We are now ready to give the algebra structure on U. Let Vi, V2, V3 be A"*",
A", Ag in some order and Vi —> [/ some embedding, invariant under Spin(8)
and preserving the inner product. Take R^ and let S3 act on it by permuting
the bsisis vectors Ci, ej, 63 as it permutes' Vi, V2, V3. Take the subspace S C R^
defined by a;i -|- X2 + X3 = 0; this is the reduced representation. Define an inner
product on R^ by (x, y) = |(a;ij/i+X2j/2+2:32/3) and let S—^ U be the inclusion
onto the two dimensional subspace invariant under E3 and preserving the inner
product. We also need the vectors which change sign under a 2-cycle: the vectors
ei — 62, 62 — 63, 63 — ei will be convenient.
Lemma 14.17. With respect to this decomposition, the invariant triJinear
/unction C/®[/®C/—>R orC has (up to a scaJar factor) the following
components :
(i) On Vi®Vj®Vk with i, j, k all different, permute the vetriables to get to
A+®A"®A' and use f.
(ii) On S®Vi®Vi take x®j/®z to ±Xi{y,z) and similarly on Vi®S®VJ
and Vi®Vi®S.
(iii) On S®S®S take x®j/®z to =F (a^iJ/iZi+2:22/222+2:32/323).

Direct treatment of F^, I 103
(iv) On all other summands, 0.
The signs ±, q= are opposite and fixed for any set of choices but can be
changed [by changing the sign of S ^-^ U).
Proof. The components of the invariant function U (QU ®U —»RorC are
certainly invariant under 5pin(8). From this it follows that they are 0 except in
f A++A- +A',
the stated cases. We have U -i—i 5 + < where, of course, U is
the 26-dimensional representation of F4.
(i) On Vi ® Vj ® Vfc we get / : A+ ® A~ ® A' -> R after permutation of
the variables (up to non-zero scalar factor). Clearly all repeated term products
vanish, and both the previous proofs that [/^ —> R is non-zero show that (A"*" -I-
A~ -I- A')^ —> R is non-zero. We may sissume without loss of generality that the
scalar is 1.
(ii) On S®Vi®Vi, x®y®z goes to ?,(a;)(j/,z) for some linear ^^ : S"->¦ R
or C, which must be invariant under the 2-cycle in S3 which interchanges Vj
and Vfc where i ^ j ^ k =^ i, so i^ must be a scalar multiple of the function
XI—>Ii. Thus we get x ® y ® z i—<-aiXi{y, z) for some scalcir Oi. Because of the
E3-symmetry, we must have ai = aj = 03 = a, say.
(iii) Similarly, representation theory shows that on S ® S ® S there is only
one Ss-invariant trilinear function up to a scalar multiple. The function given is
such a function so we have x ® y ® z 1—> P^XtytZi for some scalar /3.
i
We proceed to determine q, /3. Consider Spin{9) C F4 and first determine
a. The restriction of our invariant map to A® A® Aj is invariant under Spin{9)
and Lemma 14.16 applies to it. We see that the components agree with those
here. Let us write e^, e_ for the e^ corresponding to A"*", A", so that e± ?
A"^, e+ — e_, 2eo — e+ — e. ? 5. Thus we have:
(a+ -f a") ® F+ -I- 6-) ® (e+ - e_) 1-^ ± ((a+,6+) - (a", ft"")), by Lemma 14.16,
I—> a((a''",6+) — (a~,6")), by the above.
Thus Q = ± 1.
To determine P, note that the restriction of our invariant map to Ag ® Ag ® 1
is also invariant under Spin{9). Now, on Aj ® Aj ® 1 we have
i;® u' ® Beo - e+ - e_) i->-2a(ii,ii'), u.u' e A|.
By invariance under Spin{9), this must also hold for v,v' ? Ag. In particular,
(e+ - e-) ® (e+ - e_) ® Beo - e+ - e_) i-> 2a,
where e+ — e_ ? 1 C AJ = 1 -i- AJ. But by the above, this maps to -2/3. Thus

104 Chapter 14
We now create a 27-dimensional algebra with a unit since this is the usual
construction (rather than a 26-dimensional representation U, without \init). We
restate Lemma 14.17 in the new notation.
Theorem 14.18. Take the 27-djmensionaJ space H^ + Vi +Vi + V3 and
place on it the following structure;
(i) a linear form defined on R^ by l{x) = Xi + Xj + X3 and zero on ail other
sunini«inds;
(ii) a symmetric bilinear form b defined on R' ® R^ to be the usual inner
product, on Vi®Vi twice the usual inner product and zero elsewhere;
(iii) a symmetric form t defined on Vi®V^®Vi {'<¦?'J ?" ^ ?" ^) by permuting
the variables and then using / : A"*" ® A" ® A| —> R,
on R^®V^®Vj, efc®ii®ii'
1-ton R^®R^®R^
ei ® e,- ® et i-»
{v,v') if j = i ^ k
0 if j^i;
1 if i = j = k
0 otherwise.
On all other summands t is zero.
Then, the subspace ? = Q is isomorphic to the 26-dimensional representation
U of F4 carrying b and t onto the two structure maps up to a scalar for each.
See Theorem 16.1. (We arrange to have /3 = 1, a = —1 by, for example, choice
of 5 ^ U.)
Proof. In the proof of Lemma 14.17, we take q = —1 and x ® u ® u' i—>
li{x){v,v'), ?,(x) = —X,. But in the statement of the theorem, ii{x) — Xj + Xk-
This equals —X; on S, since Xj + xj + X3 = 0 there. Q
To make the algebra product quite explicit, let J = R^-I-Vi+Vj + Vs. Then the
product o : J X J —> J is defined by 6(xoy, z) = t[x, y, z). Compare Theorem 16.1
and the definitions preceding it.
We have seen that the crucial part of the structure of U is the trilinear map
/ and that / is connected with the existence of a normed algebra in dimension 8.
In the next chapter we review the construction of this normed algebra.

Chapter 15
The Cayley numbers
What is at the root of all the exceptional phenomena we have to study: the tri-
linear maps of Proposition 5.7, Lemma 14.16 and the skew symmetric product of
Corollary 14.7? We will see that F4 and hence the other exceptional groups are
intimately related to the algebra of Cayley numbers. In particular (Theorem 15.14),
we give a construction of the triality of Proposition 5.7, using the Cayley
numbers. We also prepare results such as Lemma 15.9 and Corollary 15.12 for use in
Chapter 16 to construct F4 and its 27-dimensional representation, independently
of the previous Chapters, see Theorem 16.1. A reference for this Chapter is [11];
see also [Fr].
Theorem 15.1. There is a unique normed algebra of dimension 8 over R.
(A normed algebra is one for which \\xy\\ = ||x|| ||j/||.)
Proof. If we just wanted to prove the existence part, we could quote
Corollary 14.15, but as we want uniqueness also, we may ?is well prove that first, as
it will lead to the construction of the Cayley numbers. Suppose we have a linear
map /i:C/®V—»W with ||/i(u®v)!! = ||u|| HuH and [/, V, W vector spaces
over R with dimV = dimW, dim[/ > 0 and ||tJo|| = 1 for some tjo ? C/. We
will write /i(u®i') as uv.
The map V —» W, u h^ uu, is an isomorphism, so without loss of generality,
we can assume C/ = R"", V = W = R" (with the usual Euclidean norm).
Identify tJo with 1 ? R"" since it acts as a unit. Assume that p > 2 and 1, i
are orthonormal in R''.
Lemma 15.2. For aiJ x ? R", i{ix) = —x.
Proof. I I—* ix is an isometry of R" so by choice of an orthonormal basis,
105

106
Chapter 15
it can be written
f cos ^1 — sin Bx
sin 61 cos ^1
We also have
cos Q% — sin 6i
sin Qi cos 6i
±17
||(a + 6z)x||^ = l|a + bi|nMp
for all a, 6 e R, so
a^(x, x) + 2a6(x, is) + 6^(ix, ix) = (a^ + ^){x, x)
and thus (x,ix) = 0, that is, each Qi = ±tt/2 and there are no blocks of ±1.
Hence the square of the isometry is —1. []
Let RP"' C R.'' be the orthogonal complement of A), the subspace generated
by 1.
Lemma 15.3. CZ(R''-') acts on R".
Proof. For all w ? R"""', we have a linear map R" —» R", x i—>• ux, giving a
linear map R""' -+ Fom(R",R"), which extends to r(R'"') -^ Hom(R",R"),
where T(R''~') is the tensor algebra. We check that w^ i—> — (u, u)l: this is
clear if u = 0; otherwise it is sufficient to check the case ||m|| = 1, which we do
from Lemma 15.2. []
Corollary 15.4. If p > 2, and l,i, j are orthonormai in R"", then i{jx) =
-j(ix) for dl x6 R".
Proof. By Lemma 15.3, Cl{B}) = H acts on R" and in H, ij = - ji. Q
Of course, this is the line of thought which shows that a normed product
R" (g) R" —» R" exists only if n = 1,2, 4 or 8. This is because R" becomes a
module over Ci(R""') and so n is divisible by a large power of 2 depending on
n — 1, which reduces things to the cases mentioned.
Assume that p = n > 2; then by choosing Vg € V with ||i;o|) = 1, we
can identify U with W and so arrange that R" has the structure of a normed
algebra with a two sided unit 1. We name this algebra A.
Corollary 15.5. If 1, i are orthonormaJ in A then for all a ? A, i{ia) =
-a and in particular i^ = — 1. If I, i, j are orthonormaJ in A then i{ja) =
— j[ia) and in particular ij = — ji.

The Cayhy numbers 107
Proof. See Lemma 15.2 and Corollary 15.4. n
Lemma 15.6. Suppose that 1, i, j are orthonormal in A. Then 1, i, j, ij
are orthonormal in A and they form a basis for a subalgebra isomorphic to H.
In particular, the subalgebra is associative.
Proof. (We must avoid using ?issociativity, which we do not have.) Since 1, i,
j are orthonormal and x i—> ix is an isometry, i, i = — 1, ij are orthonormal.
Similarly x —> xj is an isometry, 1, i orthonormal, so j, ij are orthonormal.
Thus 1, i, j, ij are orthonormal. To avoid confusion we will write fc = ij for
the product in A. Let R^ be the subspace spanned by i, j, k. By Lemma 15.3,
we have an action of C/(R^) on A. Define 9 : Cl{R^) -+ A by c >-* c ¦ I
and consider 0{k{ij)) = {k{ij)) ¦ 1 = k{ij) = {ij){ij) = -1, by Corollary 15.5.
Thus e{kij + 1) = 0, so e factors to a map 9 : Cl{R^)/{Cl{R^)){kij + 1) -* A.
The left ideal {Cl(R^)){kij+ 1) is actually a two-sided ideal since kij is central
in C/(R^) and so Cl{R^)/{Cl{R^)){kij + 1) is an algebra: in fact it is H. In
particular,
Cl{R^)/{Cl{R^)){kij + 1) = R + R'
where R is generated by 1 and R^ by i, j, k. Thus 9 is an isomorphism
onto the subspace of A with bsisis 1, i, j, ij. (Note that for a subalgebra of
A of dimension greater them 3, the argument would fail here since not all basis
elements can be written so simply.)
Claim. 9 preserves products: 9{uv) = 9{u)9{v).
Proof. Clearly the Claim is true if either u or u is 1 (as 9A) = 1), so it
is sufficient to take u,v ? R^. Then
9{uv) = [uv) ¦ I = uv,
9{uN{v) = {u-l){v-l)=uv. ?
That finishes the proof of the Lemma. []
Lemma 15.7. Suppose I, i, j, ij, I are orthonormal in A. Then i(j?) =
— {ij)(., so that if this is not zero, we do not have associativity in A.
Proof.
i{ji) = ~i{ij) by Lemma 15.5 as 1, j, I are orthonormal,
= i{ij) taking a = j and 1, i, i the orthonormal set in Lemma 15.5,
= — (ij)? taking 1, ij, ? as the orthonormal set in Lemma 15.5. Q
Theorem 15.8. Suppose A is a normed algebra with 1 o/dimension 8 over
R with 1, i, j, ij, ? orthonormal in A, and H is the subalgebra corresponding

108 Chapter 15
to 1, i, j, ij as in Lemma 15.6. Then we have an isometry H® H —> A,
(o,6) !—> o + b? and the product in A is given by {a,b){c,d) = {ac — db,da + bc).
Proof. ? is orthogonal to H and a >-* ia, o i—> ja, a ^-^ {ij)a are isometries
of A taking H to H. Thus i?, jE, {ij)i are orthogonal to H. Also a i-^ aC is an
isometry and 1, i, j, ij orthonormal, so i, ii, ji, {ij)i are orthonormal. Thus
they are an orthonormal basis for the orthogonal complement of H. We see that
1. h h ^h ^1 i^. 3^, (u)^ are an orthonormal basis for A and H ® H —> A,
(o, 6) 1—> o + 6^ is an isometry. It remains to work out the products.
Claim 1. ForaiJ o.d ? A, a{dtj = {da)i.
Proof. It is sufficient to prove the Claim for a,d ? {l,i, j, ij} and it
is trivial for o or d = 1. If o = d then a{ai) = —(. and [aa)i = —i by
Lemma 15.5. If a ^ d, then
a[dt) = —{ad)i by Lemma 15.7,
= [daji by Lemma 15.5. []
Claim 2. For c, d e H, we have {bl)c = {bc)L
Proof. The Claim is clear if c = 1, so assume h e {l,i, j,Jj}, c ? {i, j,ij}.
Then
{bt)c = — c[bi) by Lemma 15.5 since 6?, c, 1 are orthonormal,
= ~{bc)(. by Claim 1,
= {bc)L Q
Finally,
{b(.){db) = {b{db))l. = {b{bd))(. = di
for b e {l,i, j, ij}. Thus
{bi){de) = {bi){{bi){db)) = -db
by repeated use of Corollary 15.5.
From these Claims, we conclude that (o, 6)(c,d) = {ac — db, do + 6c). This
finishes the proof of Theorem 15.8 and the uniqueness statement in Theorem 15.1.
D
Now we prove the existence part of Theorem 15.1 by showing that H® H 15
a normed algebra with the expected properties.
Definition. A = HfflH with the product (a,6)(c,d) = {ac — db,da + be)
and A,0) as unit is called the algebra of Cayley numbers O. The conjugate
of {a,b) is (o, 6) = (a,—6). The +1 eigenspace of conjugation is (A,0))r of

The Cayley numbers 109
dimension 1 and called the set of real psirts, and the —1 eigenspace o{dimension 7
is called the set of pure imaginary Cayley numbers.
Note. The construction of O is a generalisation of C = R © R and H =
C ® C. The letter O comes from "octonians".
Lemma 15.9. For aij x,y&0,
(i) 5 = 1,
(ii) xy = yx,
(iii) Re{xy) = Re{yx); for pure imaginary Cayley numbers we have
Im{xy) = —Im{yx),
(iv) XX = ||x||^ • 1 = XX,
(v) {x,y) = {x,y) = Re{xy) = Re{yx).
Proof, (i) Clear.
(ii) yx = (c, —d){d, —b) = {cd — bd, —be — da),
xy = {a,b){c,d) = [ca — bd,—da — be).
(iii) For pure imaginary Cayley numbers x, y we have xy = yx = yx, so
Re{yx) = Re{xy) (from which it follows that Re{xy) = Re{yx) for all Cayley
numbers x, y) and Im{xy) = — lm[yx).
(iv) XX = (o, b){a, -b) = (oa + bb, -to + ba) = (||o|p + ||6|r)(l, 0).
(v) [x,y) = {x,y) because conjugation is an isometry Take the expression
for ||x|p and polarise it. Q
We next determine when associativity holds.
Lemma 15.10. For all x,y ? O, we have
x{xy) = {xx)y and {xy)y = x{yy).
Proof. Letting x=(o, 6), y = {c,d), we have
{a,b){{a,b){c,d)) = {a,-b){ac - db, da + bc)
= [doc — ddb + ddb + ebb, dad + bed — bed + bbd)
= {\\a\\' + \m{c,d)
= {xx)y.
For the second part, putting y for x and x for y gives y{yx) = {yy)x which
on taking conjugates leads to {xy)y = x{yy). Q

no Chapter 15
We can express these results in terms of alternative algebras.
Definition. In an algebra A, which need not be associative, the associator
of x,y,z € A is [x,y,z] = {xy)z — x[yz).
The algebra A is alternative if [x, y, z\ is an alternating function of x, y, z.
Theorem 15.11. The Cayley numbers are alternative. In detail, the
associator [x,y,z\
(i) is pure imaginary,
(ii) changes sign if you conjugate any variable,
(iii) is an alternating function of the three variables.
Proof, (ii) Clearly [x,y,z] = 0 if x = 1, i/ = 1 or z = 1, so in proving (ii),
we can assume the variable to be conjugated is pure imaginary. Thus for example
[x, y, z] = [- X, 1/, z] = - [x, y,z].
(iii) By Lemma 15.10, [x,x,y] =0 = [x,y,y],sohy {n),[x,x,y] = 0 = [x,y,y\,
and polarisation gives us [w -i- x, w + x, y] = 0 = [x, y + z, y + z] which lead
to [w,x,y] + [x,w,y] = 0, [x,y,z] + [x,z,y] = 0. That is, the associator is
alternating if you change its first two or last two variables. These trzinspositions
generate E3. For example, [x, y,z\ = - [y, x, z] = [y, z,x] = - [z, y, x].
(i) [x,i/,zl = (xi/)z-x(i/z)
= z(P) - {zy)x
= -[z,j/,xl
= [2,y,A
= - [x, y, A
by (ii),
by (iii).
D
Corollary 15.12. (i) For Cayley numbers x, y, z the real parts of {xy)z,
{yz)x, {zx)y, x{yz), y{zx), z{xy) are equal.
(ii) For pure imaginary Cayley numbers x, y, z, we have {xy,z) = {x,yz)
and it is an alternating function of its three variables.
(Compare part (ii) with Lemma 14.10.)
Proof. From Theorem 15.11 (i), we have Re[x,y,z] = 0, that is,
Re{x{yz) - {xy)z) = 0, so Re[x{yz)) = Re{{xy)z). But by Lemma 15.9,
Re{x{yz)) = Re{{yz)x) and so on.
(ii) (x,?/z) = Re{{yz)x) by Lemma 15.9 (v),
= — Re{{yz)x) since x is imaginary,
= -fle(z(x2/)) by(i),
= Re{z{xy))

The Cayley numbers HI
= {xy, z).
Also {yx,z) = —{xy,z) by Lemma 15.9 (v). []
Corollary 15.13. ||x|| ||2/|| = ||x2/||.
Proof. \\xy\\^=Re{{^){xy))
= Re{{yx){xy))
= Re{y{x{xy))) by the previous corollary,
= Re{y{{xx)y)) by Lemma 15.10,
= \\x\\'Re{yy)
= INPIMI'. D
We have thus shown that the Cayley numbers form a normed algebra. This
finishes the proof of Theorem 15.1. Q
Connection between the Cayley numbers and Lie groups.
Recall from Proposition 5.7 the trilinear function / : A""" ® A" ® A' —> R,
invariant under Spin{8). Make fixed choices of unit vectors xo, yo, zq in A+,
A", A' such that /(lo ® i/o ® zo) = 1-
Theorem 15.14. With this choice of (xq, 1/0,20I ^e can identify A+, A",
A' with the space O of Cayley numbers so as to preserve the inner product and
map Xo, yo, zq onto 1 and / onto the function Re{{xy)z) = Re{{yz)x) etc. in
Corollary 15.12 (i).
Proof. In Lemma 14.14, we transposed / to give a product x : A+® A" —>
A' satisfying /(x®i/®z) = {xxy,z) and ||xxi/|| = ||x|l \\y\\. As we have seen,
choice of Xo, yo, zq allows one to identify the three vector spaces and get a normed
algebra with 1 corresponding to Xo, yo, zq. This has to be isomorphic to the
Cayley numbers. The isomorphism preserves the inner product and the product,
and maps /(x ® 1/ ® z) onto [xy, z) = Re{{xy)z). If we alter the identification
of A' with Cayley numbers by z 1—> z, we still have zo corresponding to 1. []
Lemma 15.15. Suppose given a triple of linear maps 9 : A""" —* A+, (fi :
A" —> A", t/j : A' —> A' which preserves inner products and we have
f{6x ®(py® ipz) = f{x ® 1/ ® z).
Then there exists a unique g ? Spin(8) acting as 6 on A""", ifi on A" and
Tp on A^
Proof. The uniqueness is immediate. For the existence, we can suppose
without loss of generality that 6 = 1 for we certainly have a. g & Spin{S)
cancelling 6. When we choose a base point Xo ? A""", we get ?in action of CZ(R )

112 Chapter 15
on A' where R^ is the orthogonEil complement of Xq, and A^ has to be an
irreducible representation of C?(R^). But ip is a map of CT(R^)-modules, so it
has to be multiplication by a real scalar, which has to be ±1, since the inner
product is preserved. Similarly (p = ±1 and these signs are the same as / is
preserved (or use the identification of A" and A' under multiplication by Xq). If
both signs are — 1, multiply ff by g' € Spin{8) such that g' is 1 on A+ and
-1 on A- and A'. ?
Finally, as with the other exceptional groups, we identify G^ as a group of
automorphis ms.
Theorem 15.16. Gj = AutR(O).
Proof. By the proof of Theorem 15.14, AutR_{0) is the group of linear maps
9 : A+ —* A"*", (f : A~ —> A", ^ : A' —> A' preserving Xq, yo, zo, inner products
and /. By Lemma 15.15, this is the subgroup of Spin{?>) preserving Xo, yo,
Zq- It is sufficient to preserve two of them, say Xq and zq, since they determine
the third. The subgroup preserving Zo is Spin{7) and the restriction of A"*" to
Spin{7) is A, so we come down to the subgroup of Spin{7) preserving Xo € A.
This is Gi by definition (Theorem 5.5). Q

Chapter 16
Direct treatment of F4, II:
Jordan algebras
Definition and properties of the exceptional Jordan algebra J.
Here we complete the direct treatment of F4 begun in Chapter 14, by, for
example, identifying the 27-dimensionEd representation of Theorem 14.18 with an
algebra J ?ind F\ as the automorphisms of J, (Theorems 16.1 and 16.7). In the
last section we use these results on J to define the Cayley plzine and derive its
basic properties. There is a brief appendix on the definition of Jordan algebras,
special ?ind exceptional.
The exceptional Jordan algebra J is usually defined as the set of 3 x3 matrices
A = [aij], Ojj e O which are Hermitian: a^ = Ojj (so o« € R). The real
Ai X3 ?2
dimension of J is 27 and a typical element has the form A ¦
\ e R, li e o.
On this set J we have the following structure:
(i) a product 0 : J® J ->¦ J, A®B 1-^ Ao B = ^{AB + BA); (We check
that Ao B is Hermitian:
{AoB)ik = conjugate of 5E(o<jbj* + bijfflj*)
_ i
= 5 T.iO'jkbij + bjkaij)
i
= \ X.{o.kjbji + bkjaji)
= (A 0 5),,.)
(ii) a linear map ^ : J —> R defined by A 1—> Ai + A2 + A3 ; (Note that since
1{A) = lTr{B !-> A 0 5), this is determined by the product.)
(iii) a bihnear map 6:J®J->R, A® B >-> ^( A 0 B);
113

114
Chapter 16
(iv) a trilinear map t : J ® J ® J -> R, A® B ®C t-^ b{Ao B,C). (This is
sjrmmetric, but not obviously^)
Notes, (i) AoB = BoA.
(ii) diag{l,l,l) is a unit,
(iii) AoA=A^
(iv) i, b, t are determined by the product and conversely the product is
determined by b and t.
Theorem 16.1. If we identify R^ with the space of diagonal matrices
dioff(Ai,A2,A3)
and the space K with the space of Cayley numbers Xi (as in Tbeoiem 15.14),
then the vector space R^ + Vi+Vi + Va constructed in Theorem 14.18 is identified
with J so that the functions (., b and t correspond.
Proof. Clearly ^ = Ai + A2 + A3 corresponds. Let
(Ai I3 I2
X3 A2 ii
X2 Xi A3
B =
fj'i Vi y-i
Vi. P2 V\
y-i Vi M3
2A0B =
then
/ A,/ii + Xsys + X'iV2 A1I/3 + X3/i2 + X2S1 •^i52 + 2:32/1 + X2/.13 \
+ /ilA, +2/313+52X2 +/.ll2:3+2/3^2 +52^1 + A'lS2+2/3X1 +52'^3
^3/^1 + A253 + Xl2/2 X32/3 + A2/i2 + Il^l Xiy2 + X2V\+XitJ,3
+ S3 Al +/i2X3 +2/1X2 +S3X3 + A'2A2 + 2/lXl + ^3^2 +/.t2Xl + 2/1^3
X2/ii + xiyz + A32/2 X22/3 + XiiJ,2 + Asj^i 12S2 + Xi2/i + A3P3
V +2/2^1 +5li3 + M3X2 +2/2X3+SlA2+/i3:ri +2/2X2 +^1X1 +/i3A3 y
Using Lemma 15.9(v), we see that the diagonal terms of A 0 B are Ai/ij +
(x2,2/2) + (x3,2/3), A2/i2 +A3,2/3)+ (xi,2/0 and A3/i3 + (xi,2/i) + (x2,2/2). Thus
l{AoB) = ((Ai,A2,A3),(/ii,/i2,A'2)) +2(xi,2/i)+2(x2,2/2) + 2(x3,2/3)
as in Theorem 14.18, so that b corresponds.
The off diagonal entries of A 0 B are
~Xi
X2
5 (P2 + /^3)xi + 5 (A2 + h)yi + 5X22/3 + 2/2X3,
5 (/^3 + /^l)X2 + 5 (-^3 + -^1J/2 + \ X32/1 + 2/3X1,
5 (/^l + A'2)X3 + 5 (Al + A2)t/3 + 5 X12/2+2/1X2.

Direct treatment of F4, II: Jordem algebras 215
/ Ui Zi Z2 \
Now let C = Z3 1/2 Zi , and consider
\ Z2 zi 1/3 /
t{A,B,C) = b{AoB,C)
= Ai/iii/] + {xi,yi){i/2 + i^) + {xi,zi){fi2 + fii) + B/1, zi)(A2 + A3)
+ X2fl2l^2 + {X2,y2){l^3 + I'l) + {X2, Z2)(rt + Ml) + B/2, Z2)(A3 + Aj)
+ A3/i3l'3 + (X3,2/3)A'1 + 1^2) + (l3,Z3)(/il + M2) + B/3,Z3)(Ai + A2)
+ Re{{x2Vi + 2/2i3)zi + B:32/1 + 2/32:i)z2 + A12/2 + yiX2)Zi).
Using Corollary 15.12(i), we see that the last line is
fle((Xi2/2)z3 + {yiX2)Zi + B/lZ2)X3 + (Zi2/2)X3 + (XlZ2J/3 + (ZlX2J/3)-
Finally, using the identification of the trilinear function A+ ® A~ ® X' —>
R with Re{xy)z) given in Theorem 15.14, our t corresponds exactly to the t in
Theorem 14.18 on F4 and it is also now clear that the function t is symmetric.
D
Corollary 16.2. The 26-dimensionai representation U of F4 is isomorphic
to the subspace i = 0 of J in such a way that the products correspond up to
scaJars.
Proof. This follows from Theorem 16.1 and Theorem 14.18. Q
Corollary 16.3. The group F4 acts on J as a group of automorphisms
preserving the product.
Proof. Let F4 act so as to fix 1 and act on the subspace i = 0 as it does on
U. It then preserves i and it preserves b,t on U®U, U®U®U. Therefore F4
preserves the component of the product from U®U into f/, but also from U ®U
to Rl, since this is determined by l[A 0 B) = b{A, B) ?ind b is preserved, so it
preserves products on U <^U. But certainly, it preserves products by 1, because
1 acts as a unit, so it preserves all products in J. []
So we have a monomorphism F4 —* Aut{J).
Proposition 16.4. Any element of J can be mapped by an element of Ft
into diagonal form diag{Xi, A2, A3).
(Ai I3 X2 \
?3 A2 ii in an Ft-orbit. The
X2 X\ A3 /
function ||2:i||^+||2:2|i^+||i3|P is continuous on this orbit and the orbit is compact

116 Chapter 16
because F4 is compact. Therefore the function attains its minimum in the orbit.
Suppose, for a contradiction, that the minimum is strictly greater than zero and
attained at some element of the orbit with X] 5^ 0. F4 contains three copies of
Spin{9), one for each of Vl, V2, V3. The one corresponding to Vi acts according
to the representation Ag on the 9-dimensional space spanned by Vi and matrices
diag{0, A, -A). It acts by A on V2 + V3, the space of the X2 and X3. Spin[9) is
transitive on vectors of a fixed length in Ag, so by using an element of Spin[9),
we can replace our representative by one with a new A2, A3 and new Xj = 0.
This does not change ili2|P + ll^^all^, so it diminishes ||ii||' + 11^21^ + llxalp,
contradicting our assumption that our representative attained the minimum value.
D
Addendum 16.5. A diagonal matrix diag{Xi, A2, A3) can have its diagonal
entries permuted by elements o{ F\.
Proof. By using an element of 5pin(9) (C F4) we can preserve diag{l, 0,0)
and rfio5@,l,l) and reverse diag((l,\,—X). That is, we can interchange A2, A3.
Similarly we get the other transpositions and hence all of E3. Q
Theorem 16.6. (i)
The aJgebra of polynomial functions on J invariant under Aut(J)
= the algebra of polynomial functions on J invariant under F4
= the polynomial algebra generated by the functions i{A), b{A, A), t{A,A,A).
(ii) An element of J is determined up to action of F^ by the values of i{A),
b{A,A) and t{A,A,A) there.
(iii) Any element of J satisfies a cubic equation
A^ -CiA^ + C2A-C3I = 0,
where A'= Ao A, A^ = (A0 A) 0 A = A0 (A0 A), ci = e{A), Cj = |(^(AJ -
6(A,A)) and C3 = i(^(A)'-3^(AN(A, A) + 2t(A, A, A)). (Note that the product
0 is commutative, but not associative!)
Proof, (i) Call the three algebras Pj, P2.-fa, respectively. Already we have
P3 C Pi C Pi- A polynomial function of J, invariant under F4, is determined by
its values on diagonal matrices diag{X,fi, u), by Proposition 16.4. The value is a
polynomial in A, p, u and symmetric by Addendum 16.5. Thus it is a polynomial
in the elementary symmetric functions or in the power sums X+fj,+u, X^+fi^ + u^,
A^+ /i^+1/^, which are the restrictions of i{A}, b{A,A), t{A,A,A) respectively
to diagonal matrices. Hence P2 C P3 and (i) is proved.
(iii) It is sufficient to check for a real matrix A = diag{X,fi,i/). For such a
matrix we get A'-CiA^ + C2A-C3 = 0, where Ci = X + fi + v, c^ = Xfi + fiu + uX,
C3 = Xfii'.

Direct treatment of F4, II: Jordan algebras 117
(ii) Any element of J can be mapped by F4 onto diag{X,fi,u) by
Proposition 16.4. The functions i{A), b{A,A), t{A,A,A) determine A, fj,, v up to
order (by part (iii)) which determines diag[X,fi,u) up to the action of Fi by
Addendum 16.5. Q
Theorem 16.7. (i) F4 = Aut{J) hy the above action.
(ii) The subgroup of Aut{J) Bxing ei = diag{l,0,0) is Spin{9).
(iii) The subgroup of Aut{J) Bxing ei, 62 = diag{0,l,0) and 63 =
diag{0,0,l) is Spin{S).
Proof, (iii) Consider the hnear map J -^ J given by A i~* ei 0 A. Its 1-
eigenspace is 1-dimensional, generated by ei. Its 0-eigenspace is 10-dimensionai,
generated by 62, 63 and Vi, the space of matrices in which the first row and
column are zero. Its ^-eigenspace, of dimension 16, is V2 + V3, the space of
/ 0 X3 12 \
matrices of the form X3 0 0 If 5 : J —<• J fixes
\X2 0 0 y
ei, it preserves V2 + V3,
62, it preserves Vz + Vi,
63, it preserves Vi + V2.
Thus, if 9 preserves ei, 62 and 63, it preserves Vi, V2 and V3 separately and
such a 9 consists of linear maps 1 : R' —* R^, K —> Vi, preserving inner products
and /. Thus 9 lies in Spin{S) by Lemma 15.15.
(ii) Let 9 : J -^ J preserve ei. It also preserves 1 = ei + 62 4- 63, therefore
preserves 62 + 63. Now e\ 0 [a — 63) = 0, so 9{ei o (e2 — 63)) = 9ei 0 9{ei — 63) =
ei 0 9{e2 — 63) = 0. Thus ^(ej — 63) is in the 0-eigenspace of A 1—> ei 0 A. We
/ 0 0 0 \
conclude that 5 F2 —63) = 0 A xi ., where A^-|-Xiii = 2. So there exists
V 0 xi -X J
g e Spin{9) such that 5F2 — 63) = 9{e2 — 63) and we can reduce the problem
to the case of </> = g'^9 fixing ei, 62, 63. Thus (^ = g~^9 ? 5pin(8) by (iii) and
9 e Spin(9).
(i) Suppose 9 : J —* J is an automorphism ?ind consider 9ei. This is a
point A with ^(A) = 1 = 6(A, A) = t(A, A, A). By Theorem 16.6(ii), there exists
5 e F4 such that g~^9ei = e\. Thus we can reduce the problem to the case of ?in
automorphism (j> = g~^9 which preserves ei; such a </> must lie in Spin{9) by
part (ii). Hence S ? F4. Q

118 Chapter 16
The Cayley projective plane.
In constructing this plane, our problem is, of course, the lack of
associativity. We try to learn from the case of H. A point of P^(H) is a 1-dimensional
submodule C H^ ?ind this can be identified with a projection operator, namely
"orthogonal projection onto this 1-dimensional subspace", which is a 3 x 3 Her-
mitian matrix, A, over H, idempotent (A^ = A) and of rank 1.
In the exceptional Jordan algebra of 3 X 3 matrices over O, let 11 be the
F4-orbit of ei = diag{l,0,Q), that is, the space of all A such that i{A) = 1,
b{A, A) = 1, t{A, A, A) = 1: such A do satisfy A^ = A and have rank 1. Hence
n = Fi/Spin{9) is a Cayley projective plane. (Cf. Theorem 16.7 (ii).) Recall
from the proof of Theorem 16.7 that ei, acting on J by o, has a
1 eigenspace of dimension 1 generated by ei;
I eigenspace of dimension 16, the tangent space, V2 + V3, to II at ei;
0 eigenspace of dimension 10, generated by 62, 63 and Vi.
The points of II in the 0 eigenspace, i.e. the rank 1 idempotents orthogonal to
ei, form an 5' with Spin{9) acting on it — the space of points in II which
attain maximum distance from ei. They constitute a Cayley line in II. (See [He,
p. 167].) As befits a projective plane, two distinct points of H determine a unique
Cayley line joining them (and two distinct Cayley lines meet in a unique point
of n). The condition for three points to be collinear is the following. Take the
function Xfiu corresponding to C3 = ^(^(A)^-3^(AN(A, A)-l-2t(A, A, A)), and
polarise to get the associated symmetric trilinear function
i{A)e{B)i{C) - C{A)b{B, C) - e{B)b{C, A) - C{C)b{A, B) + 2t(A, B, C).
This function is > 0 on H and is equal to zero if and only if A, B, C are
collinear. Finally, by transposing this trilinear function, we get
A X B = 2A 0 5 - ?(AM - i{B)A + {i{A)i{B) - b{A, B))I,
a point on the Cayley projective plane determining the equation of the line joining
A and B, viz., the set of points which attain maximum distance from A x B or
are orthogonal to Ax B.

Direct treatment of F4, 11: Jordan algebras 119
Appendix. Jordan algebras
Definition. A special Jordan algebra is a subset of an associative algebra
which is closed under the new product Ao B = |(A5 + BA).
Examples. In M„(F) with F = R, C or H, take the subset of Hermitian
matrices.
Note. The class of special Jordan algebras cannot be characterised by
algebraic identities.
Proof. There is a homomorphism from a special Jordan algebra onto an
algebra which is not a special Jordan algebra. For details see [Co]. Q
In a special Jordan algebra, we have the following
Proposition, (i) {Ao B) o [Ao A) = Ao {B 0 [Ao A)).
(ii) The associator, as a function of its middle variables, is a derivation with
respect to 0, that is [A,B oC, D] = [A,B,D]o C + B 0 [A, C, D].
Proof, (i) {AoB)o{AoA)
= ^(AB + 5A)oA2
= \{A^B + ABA'' + A^BA+ BA^)
= \{A^B + A'BA + ABA"" + BA^)
= \{A{BA^ + A^B) + {BA" + A''B)A).
(ii) [A,X,D]
= {AoX)oD-Ao{XoD)
= I {{AX + XA)oD-Ao {XD + DX))
= \{AXD + XAD + DAX + DXA - AXD - ADX - XDA - DXA)
= \{[X,AD\-[X,DA])
= \{[[D,A\,X]).
But in any associative algebra,
[E,XY] =EXY-XYE
= EXY - XEY 4- XEY - XYE
= [E,X\Y+X[E,Y].
that is, the commutator is a derivation as a function of the second variable. So,
[?, XoY\ ={{[E, X]Y + X[E, Y] + [E, Y]X + X[E, Y])
= [E,X]oY + Xo[E,Y]. Q

120 Chapter 16
In fact in a special Jordan algebra, (i) => (ii), but (ii) 7^ (i). The classical
definition of a Jordan algebra is an algebra with a commutative product 0, such
that (i) (A 0 B) 0 (A 0 A) = A 0 E 0 (A o A)) holds.
A Jordan algebra which is not special is called exceptional. For example the
algebra J of Chapter 16 is exceptional.

References
[I] J. F. Adams, Lectures on Lie groups, Benjamin, 1969; reprint by the
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[Ad 1] J. F. Adams, 2-Tori in Es, Math. Ann., 287 A987), pp. 29-39.
[Ad 2] J. F. Adams, The fundamental representations of Es, Contemp. Math., 37
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[Al] A. A. Albert, A structure theory for Jordan algebras, Ann. of Math., 48
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122 REFERENCES
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