W H AT I S . . . G 2 ?
MARTIN H. WEISSMAN
MARCH 17, 2009

Octonions
Let k be a field, with char (k) 6= 2. For example, k may be Q or Ror C (though the last case is not as interesting). A Hurwitz algebra over k is a finite-dimensional, k-algebra A with unit element,
together with a quadratic form N : A ! k, such that the associatedbilinear form is nondegenerate and:N ( xy) = N ( x ) N (y), for all x, y 2 A.

Every Hurwitz algebra1 over k has dimension 1, 2, 4, or 8, as a kvector space.
In what follows, we write k, K, B, C for Hurwitz algebras ofdimensions 1, 2, 4, 8 respectively; K is called a quadratic étale kalgebra, B is called a quaternion algebra over k, and C is called aCayley or octonion algebra over k.
Hurwitz algebras become more pathological as their dimension
rises: quadratic étale algebras are commutative and associative.
However, quaternion algebras are associative but never commutative. Cayley algebras are neither commutative nor associative.
However, Cayley algebras are alternative, so that for any two elements x, y 2 C, the algebra generated by x and y is associative(e.g., ( xy) x = x (yx )).
There are exactly two complete chains of Hurwitz algebras overthe real numbers R. First is the sequence R ⇢ C ⇢ H ⇢ O, where,
C denotes the complex numbers, H Hamilton’s quaternions, andO Graves’s octonions. 2 The other complete chain of Hurwitz algebras over R is R ⇢ R ⇥ R ⇢ M2 (R ) ⇢ O spl , where R ⇥ R
denotes the algebra of ordered pairs of real numbers (multiplication entry-wise), M2 (R ) denotes the two-by-two matrix algebra,and O spl denotes the “split octonions”, which we discuss next.
For any field k (or even a commutative ring!), Zorn’s algebraof split octonions over k is defined to be the set of all two-by-two“matrices”:✓◆
a ~v~ 2 k 3 },w={
: a, d 2 k, ~v, w~ dw
with composition given by the following formula:3✓◆ ✓◆ ✓◆~~~ ⇥ya ~v
a ~f
aa + ~v · ya~f + d~v w.
· ~=~ + ~v ⇥ ~f~ dw~ + dy~ · ~fy daw
dd + w

Resulting structures on a Hurwitz algebra A
include a trace,
Tr ( a) = N ( a + 1)

N ( a)

N (1),and involution ā = Tr ( a) a. Hurwitz
algebras are quadratic; every element asatisfies the polynomiala2

Tr ( a) + N ( a) = 0.

1This was first proven over R by Adolph
Hurwitz, in Über die Composition der quadratischen Formen von beliebig vielen Variablen,found in Math. Werke I. For a general field,
a proof can be found in Irving Kaplansky,
Infinite-dimensional quadratic forms admitting
composition, Proc. Amer. Math. Soc. 4, (1953).

Figure 1: A. Hurwitz (1859-1919), conducting,while his daughter plays violin with A.
Einstein. Taken from David Rowe’sIntelligencer article Felix Klein, Adolph
Hurwitz, and the “Jewish Question” in German
academia (original source, Polya’s PhotoAlbum).
2

A nice historical overview of the quaternions and octonions can be found in the firstfew pages of Baez’s article The Octonions, in
Bull. A.M.S. Vol. 39, 2001.
Max Zorn, 1906–1993, a student of Artin,
most famous probably for “Zorn’s Lemma”,
but also active in algebra and analysis. His
work on “octonions” can be found in Theorie
der Alternativen Ringe, Abh. Math. Sem.
Hamburgishcen Univ., vol. 8 (1930).
3Here, we use the standard dot product,cross product, and scalar multiplication.

1

what is... G2 ?

2Automorphisms and Embeddings
Let us fix a base field k (char (k ) 6= 2), and a complete chain4 of
Hurwitz algebras k ⇢ K ⇢ B ⇢ C. One may consider the groups of
k-linear automorphisms that preserve the algebra structure5 ; thisyields groups:

4It would not be too harmful to consider thechainR ⇢ R ⇥ R ⇢ M2 (R ) ⇢ O spl .
5Aut(k/k ), Aut(K/k ), Aut( B/k), Aut(C/k ).
The group Aut(k/k) is trivial, and the automorphism Aut(K/k) ={1, s}, where s( a) = ā for all a 2 K. The groups Aut( B/k) andAut(C/k) are more complicated.
Suppose, for example, that B = M2 (k ) is the “split quaternion
algebra”6 . If g 2 GL2 (k), then define the inner automorphism
Int[ g] by:Int[ g](b) = gbg 1 , for all b 2 B.
Then, Int is a homomorphism from GL2 (k ) to Aut( B/k). Furthermore, if z 2 Z ( GL2 (k)) is a scalar matrix, then Int[z] = 1.
Conversely, if g 2 GL2 (k), and Int[ g] = 1, then g commuteswith every b 2 B, and hence g 2 Z ( GL2 (k)). It follows that Int
descends to a unique injective homomorphism:

Since every element a of a Hurwitz algebra
satisfies a quadratic identity a2 Tr ( a) +N ( a) = 0, it can be seen that the algebra
structure determines the quadratic form N,
the trace map Tr, and hence the involutionā = Tr ( a) a.

6The same methods work for nonsplit quaternion algebras as well; every automorphismof a quaternion algebra is inner, yielding anisomorphism from B⇥ /Z ( B⇥ ) to Aut( B/k ).
Indeed, every quaternion algebra splits overa Galois extension, and so descent applies toprove that Int is an isomorphism.

Int : PGL2 (k) = GL2 (k)/k⇥ ,! Aut( B/k).
Conversely, if a 2 Aut( B/k), then a is determined by the resulting linear automorphism a of the trace-zero subspace B ⇢ B.
Moreover, a preserves the norm quadratic form N on B , so thatAut( B/k ) embeds into O( B , N ). In fact, a bit more work7 shows
that:PGL2 (k) ⇠= Aut( B/k) ⇠= SO( B , N ).
Automorphisms of Cayley algebras are more complicatedthan automorphisms of quaternion algebras. An automorphism
a of (for example, the split) Cayley algebra C restricts to a linear,norm-preserving, automorphism a of the seven-dimensional
trace-zero subspace C . Furthermore, a is determined from a .
This provides an embedding Aut(C/k) ,! O(C , N ).
The resulting group Aut(C/k ) is not easy to describe, but has
another name:
G2,C = Aut(C/k).
The answer to the title of this lecture is: G2 is the automorphismgroup of a Cayley (a.k.a. octonion algebra); which Cayley algebra
and which base field should be made clear.

7
To show that the image of Aut( B/k ) inO( B , N ) is contained in SO( B , N ), onemust check that no automorphism of B acts
as a reflection of B . A reflection would
preserve two orthogonal vectors x, y in B ,and send a third vector z to its negative.
But, a basis of B as a k-vector space is given
by x, y, ( xy ȳ x̄ ), so that the action of analgebra automorphism a on B is uniquelydetermined by its action on x and y. In
particular, if a fixes x and y, then a fixes z.
This demonstrates that Aut( B/k ) injectively
maps to SO( B , N ). For surjectivity, adimension argument suffices.

what is... G2 ?

3

Subgroups of Automorphisms
Given a complete chain k ⇢ K ⇢ B ⇢ C of Hurwitz algebras overk, we have considered groups:
Aut(k/k ), Aut(K/k), Aut( B/k), Aut(C/k ),each arguably more interesting than the previous. The only
group which is “really new” is Aut(C/k). To “know” the group
Aut(C/k), it is most helpful to understand its subgroups8 . Two
subgroups which arise most generally are Aut(C/K ) and Aut(C/B):
the algebra automorphisms of C, which fix every element of K orof B, respectively.
First, let us consider a 2 Aut(C/K ); such an automorphism apreserves every element of the k-subspace K, and preserves thenorm and trace, and hence stabilizes the subspace K ? of elements
orthogonal to K:
K ? = {w 2 C : Tr (w z̄) = 0 for all z 2 K }.

The alternative property9 of C implies that K ? is a (left) K-module:in particular,
w 2 K ? , z 2 K ) z · w.

Now, every element of C can be expressed uniquely as a sum
z + w (or ordered pair (z, w )), where z 2 K and w 2 K ? . There is aunique Hermitian form10 F : K ? ⇥ K ? ! K, such that:
projK (z, w ) · (z0 , w 0 ) = (zz0F(w, w 0 )).

8A good reference for automorphisms ofCayley algebras is Jacobson’s Composition
algebras and their automorphisms, Rend. Circ.
Mat. Palermo 1958

9The nontrivial fact is that if z, w 2 K, andw 2 C, then z · (w · w ) = (z · w) · w. But thealternative property implies that such a triple
z, w, w lie in an associative subalgebra of C.

10For all z, z0 2 K, and w, w 0 2 K ? ,F(zw, z0 w 0 ) = zz̄0 F(w, w 0 ).

It follows that every automorphism a 2 Aut(C/K ) preserves
this Hermitian form. More precisely, one arrives at an injective
homomorphism Aut(C/K ) ,! U (K ? , F). The most precise possible result is that, in fact, Aut(C/K ) is isomorphic to the group
SU (K ? , F).
A very good example of Aut(C/K ) is provided by Zorn’s split
octonions, and the algebra K = k ⇥ k embedded as the diagonal
matrices. Then we find that:✓◆0 ~v~ 2 k 3 }.
K? = {: ~v, w~ 0w

One can verify that SU (K ? , F) is isomorphic to SL3 (k). Specifically, if g 2 SL3 (k), then the action of g on C is given by:
✓◆ ✓◆
ag~v
a ~v
g= t 1.
~ dw~ dg w
This yields the long root embeddingelong : SL3 (k ) ,! G2,C .

Here, we identify K ? with k3 ⇥ k3 , with K3 ,when K = k ⇥ k. Note that “conjugation” in
K is given by z̄ = (y, x ) if z = ( x, y). If ~z =~ = ( w1 , w2 , w3 ) 2 K 3 ,(z1 , z2 , z3 ) 2 K3 and w
then the Hermitian form is given by:~ ) = z1 w̄1 + z2 w̄2 + z3 w̄3 .
F(~z, wwhat is... G2 ?

4Quaternion subalgebras
One may also consider the automorphisms Aut(C/B) fixing every
element of the quaternion algebra B. Since K ⇢ B, we find thatAut(C/B) ⇢ Aut(C/K ). As in the case of quadratic subalgebras,
we may consider the orthogonal complement B? of B in C. Thereexists an element ` 2 B? , such that every element of C has theform a + b`, for some a, b 2 B. Moreover, if a 2 Aut(C/B), thenit is a theorem that there exists a unique element ua 2 B such that
N (ua ) = 1 anda( a + b`) = a + (ua b)`, for all a, b 2 B.

highest longThis provides an isomorphism:Aut(C/B) ⇠= SB⇥ = {u 2 B⇥ : N (u) = 1}.

When B = M2 (k ), SB⇥ = SL2 (k ), providing the highest root
embedding ehigh : SL2 (k ) ,! G2,C . Of course, since Aut(C/B) ⇢Aut(C/K ), the image of ehigh is contained in the image of elong .
Another embedding of SB⇥ into G2,C follows: if g 2 SB⇥ , anda + b` 2 C, define:longshort

longg( a + b`) = ( gag 1 ) + (bg 1 )`.

This defines the short root embedding eshort : SL2 (k) ,! G2,C .
Observe that the images of ehigh and eshort commute11 , and theirintersection consists of {±1} ⇢ SB⇥ . This provides the embedding:12ehigh ⇥ eshort : SB⇥ ⇥±1 SB⇥ ,! G2,C .
Suppose C is Zorn’s split Cayley algebra. If t is a cyclic permutation of {1, 2, 3}, then t naturally acts (by permuting basiselements) on k3 . This yields an automorphism:✓◆ ✓◆a ~v
a t~v=.
t~ d~ dw
twWe find an embedding w : A3 ,! G2,C in this way.
There are three ways of embedding M2 (k) into the split Cayley
algebra C, via:✓◆ ✓◆a ba bei
ii=,c dcei dwhere (e1 , e2 , e3 ) is the standard basis of k3 . Altogether, we find
three conjugate embeddings:
eshort,i : SL2 (k) ,! G2,C .
In fact, G2,C is generated by the images:
eshort,i (SL2 (k )), elong (SL3 (k)).

long

long

long
Figure 2: The long roots for G2 . The other six
intersection points are short roots.
11

This reflects the orthogonality of the shortroot and highest long root in the diagram
above
12

This embedding is very important for thesplit real Lie group G2 , where B is Hamilton’squaternion algebra and C is the split octonion
algebra (such embeddings exist!). There,
one finds an embedding SU2 ⇥±1 SU2 as themaximal compact subgroup of the split Liegroup G2 . From this, one finds that the split
G2 is not simply-connected as a topologicalgroup; rather, it has a two-fold covering
which is simply-connected. This leads togreat confusion!