Proof : This is a classical result of the theory of quadratic forms, due to A. P¬ster

(see for example Scharlau [?, p. 90]). As we already pointed out, the implication

(??) ’ (??) follows from Proposition (??). We include a proof of the converse

which is much in the spirit of this chapter, however we assume that char F = 2.

The idea is to construct a Cayley algebra structure on V such that the corresponding

norm is a multiple of q. This construction is similar to the construction given in

Chevalley [?, Chap. IV] for forms of maximal index. Let

∼

± : C(q), „ ’ EndF (U ), σk

’

be an isomorphism of algebras with involution where „ is the involution of C(V, q)

which is the identity on V . Let be a nontrivial idempotent generating the center

of C0 (q). By putting U1 = ±( )U and U2 = ±(1 ’ )U , we obtain a decomposition

(U, k) = (U1 , q1 ) ⊥ (U2 , q2 )

such that ± is an isomorphism of graded algebras where EndF (U1 • U2 ) is “checker-

board” graded. For any x ∈ V , let

0 ρx

±(x) = ∈ EndF (U1 • U2 )

»x 0

so that »x ∈ HomF (U2 , U1 ) and ρx ∈ HomF (U1 , U2 ) are such that »x —¦ρx = q(x)·1U1

and ρx —¦»x = q(x)·1U2 . Let ˆi : Ui ’ Ui— be the adjoints of qi , i.e., the isomorphisms

∼

b ’

induced by bi = bqi . We have

’1 t

ˆ1 ˆ1

b 0 0 ρx b 0 0 ρx

=

ˆ2 ˆ2

»x 0 »x 0

0 b 0 b

hence

ˆ1 —¦ ρx = »t —¦ ˆ2 and ˆ2 —¦ »x = ρt —¦ ˆ1

b xb b xb

or, putting ρx (u2 ) = ρ(x, u2 ) and »x (u1 ) = »(x, u1 ), we obtain maps

» : V — U 1 ’ U2 and ρ : V — U2 ’ U1

such that

b1 ρ(x, u2 ), u1 = b2 u2 , »(x, u1 ) .

If we set u2 = »(x, u1 ) we then have

b2 »(x, u1 ), »(x, u1 ) = b1 bq (x)u1 , u1

so that, since we are assuming that char F = 2,

q2 »(x, u1 ) = q(x)q1 (u1 )

§35. CLIFFORD ALGEBRAS AND TRIALITY 481

for x ∈ V and u1 ∈ U1 . Similarly the equation q1 ρ(x, u2 ) = q(x)q2 (u2 ) holds for

x ∈ V and u2 ∈ U2 . By linearizing the ¬rst formula, we obtain

b2 »(x, u1 ), »(x, v1 ) = q(x)b1 (u1 , v1 )

and

b2 »(x, u), »(y, u) = bq (x, y)q1 (u)

for x, y ∈ V and u1 , v1 ∈ U1 . By replacing q by a multiple, we may assume that

q represents 1, say q(e) = 1. We may do the same for q1 , say q1 (e1 ) = 1. We then

have q2 (e2 ) = 1 for e2 = »(e, e1 ). We claim that ρ(e, e2 ) = e1 . For any u1 ∈ U1 , we

have

b1 ρ(e, e2 ), u1 = b2 e2 , µ(e, u1 )

= b2 »(e, e1 ), »(e, u1 )

= b1 (e1 , u1 )q(e) = b1 (e1 , u1 ).

Since q1 is nonsingular, ρ(e, e2 ) equals e1 as claimed. The maps s1 : V ’ U1 and

s2 : V ’ U2 given by s1 (x) = ρ(x, e2 ) and s2 (y) = »(y, e1 ) are clearly isometries.

Let

x y = s’1 » x, s1 (y) for x, y ∈ V .

2

We have

q(x y) = q2 » x, s1 (y) = q(x)q1 s1 (y) = q(x)q(y),

x e = s’1 » x, s1 (e) = s’1 »(x, e1 ) = x

2 2

and

bq (v, e y) = b2 »(v, e1 ), µ(x, e1 ) = bq (v, e)

for all v ∈ V , so that e y = y. Thus V is a composition algebra with identity

element e. By Theorem (??) (V, ) is a Cayley algebra.

As an application we give another proof of a classical result of the theory of

quadratic forms (see for example Scharlau [?, p. 89]).

(35.3) Corollary. Let (V, q) be an 8-dimensional quadratic space with trivial dis-

criminant and trivial Cli¬ord invariant. Then (V, q) is hyperbolic if and only if it

is isotropic.

Proof : By Proposition (??), q is similar to the norm of a Cayley algebra, so that

the claim follows from Proposition (??).

35.B. Similitudes and triality. Let (S, , n) be a symmetric composition

algebra of dimension 8 over F . In view of Proposition (??) any similitude t of n

induces an automorphism C0 (t) of C0 (n) and t is proper, resp. improper if C0 (t) re-

stricts to the identity of the center Z of C0 (n), resp. the nontrivial F -automorphism

ι of Z. Let GO+ (n) be the group scheme of proper similitudes of n and GO’ (n)

the set of improper similitudes. The “triality principle” for similitudes of (S, n) is

the following result:

482 VIII. COMPOSITION AND TRIALITY

(35.4) Proposition. Let t be a proper similitude of (S, n) with multiplier µ(t).

There exist proper similitudes t+ , t’ of (S, n) such that

µ(t+ )’1 t+ (x y) = t(x) t’ (y),

(1)

µ(t)’1 t(x y) = t’ (x) t+ (y)

(2)

and

µ(t’ )’1 t’ (x y) = t+ (x) t(y).

(3)

Let t be an improper similitude with multiplier µ(t). There exist improper simili-

tudes t+ , t’ such that

µ(t+ )’1 t+ (x y) = t(y) t’ (x),

(4)

µ(t)’1 t(x y) = t’ (y) t+ (x)

(5)

and

µ(t’ )’1 t’ (x y) = t+ (y) t(x).

(6)

The pair (t+ , t’ ) is determined by t up to a factor (µ, µ’1 ), µ ∈ F — , and we have

µ(t+ )µ(t)µ(t’ ) = 1.

Furthermore, any of the formulas (??) to (??) (resp. (??) to (??)) implies the

others. If t is in O+ (n), the spinor norm Sn(t) of t is the class in F — /F —2 of the

multiplier of t+ (or t’ ).

Proof : Let t be a proper similitude with multiplier µ(t). The map S ’ EndF (S•S)

given by

0 1 0

t(x)

•(t) : x ’ = ±S t(x)

’1

0 µ(t)’1

µ(t) rt(x) 0

is such that (•(t)(x))2 = µ(t)’1 n t(x) = n(x), and so it induces a homomorphism

∼

•(t) : C(n) ’ EndF (S • S).

’

By dimension count •(t) is an isomorphism. By the Skolem-Noether Theorem, the

s0 s1

automorphism •(t) —¦ ±’1 of EndF (S • S) is inner. Let •(t) —¦ ±’1 = Int s3 s2 .

S S

Computing ±’1 —¦ •(t) on a product xy for x, y ∈ S shows that ±’1 —¦ •(t)|C0 =

S S

C0 (t). Since t is proper, C0 (t) is Z-linear. Again by Skolem-Noether we may write

s0 0

±S —¦ C0 (t) —¦ ±’1 = Int 0 s . This implies s1 = s3 = 0 and we may choose s0 = s0 ,

S 2

s0 0

s2 = s2 . We deduce from •(t)(x) = Int —¦ ±S (x) that

0 s2

= s0 x s’1 and µ(t)’1 rt(x) = s2 rx s’1

t(x) 2 0

or

s0 (x y) = t(x) s2 (y) and s2 (y x) = µ(t)’1 s0 (y) t(x), x, y ∈ S.

The fact that C0 (t) commutes with the involution „ of C0 (n) implies that s0 , s2 are

similitudes and we have µ(s0 ) = µ(t)µ(s2 ). Putting t+ = µ(s0 )’1 s0 and t’ = s2

gives (??) and (??).

To obtain (??), we replace x by y x in (??). We have

µ(t+ )’1 n(y)t+ (x) = t(y x) t’ (y).

Multiplying with t’ (y) on the left gives

µ(t+ )’1 n(y)t’ (y) t+ (x) = t(y x)µ(t’ )n(y).

§35. CLIFFORD ALGEBRAS AND TRIALITY 483

By viewing y as “generic”(apply (??)), we may divide both sides by n(y). This

gives (??).

If t is improper, then C0 (t) switches the two factors of Z = F — F and, given t,

we get t+ , t’ such that µ(t+ )t+ (x y) = t(y) t’ (x).

Formulas (??) and (??) follow similarly.

Conversely, if t satis¬es (??), then C0 (t) switches the two factors of Z = F — F ,

hence is not proper. This remark and the above formulas then show that t+ , t’ are

proper if t is proper. To show uniqueness of t+ , t’ up to a unit, we ¬rst observe

that t+ , t’ are unique up to a pair (r + , r’ ) of scalars, since

t+ 0

±S C0 (t)±’1 = Int .

S 0 t’

Replacing (t+ , t’ ) by (r+ t+ , r’ t’ ) gives

µ(t+ )(r+ )’1 t+ (x y) = r’ t’ (x) t(y) = µ(t+ )’1 r’ t+ (x y).

’1

This implies r+ = r’ . We ¬nally check that Sn(t) is the multiplier of t+ (or t’ )

for t ∈ O+ (n). The transpose of a linear map t is denoted by t— . Putting ±S (c) =

(t+ , t’ ) and writing ˆ : S ’ S — for the isomorphism induced by bn , we have

b

— — — —

±S c„ (c) = (t+ˆ’1 t+ ˆ t’ˆ’1 t’ ˆ = (ˆ’1 t+ ˆ + , ˆ’1 t’ ˆ ’ )

b b, b b) b bt b bt

— —

(since c„ (c) ∈ F ). Then the claim follows from t+ ˆ + = µ(t+ )ˆ and t’ ˆ ’ =

bt b bt

’ˆ ’1 —2 — —2

µ(t )b, since Sn(t) = Sn(t ) = c„ (c)F ∈ F /F . The other claims can be

checked by similar computations.

(35.5) Corollary. For any pair », »+ ∈ D(n), the set of values represented by n,

there exists a triple of similitudes t, t+ , t’ such that Proposition (??) holds and

such that », »+ are the multipliers of t, resp. t+ .

Proof : Given » ∈ D(n), let t be a similitude with multiplier », for example t(x) =

u x with n(u) = », and let t+ , t’ be given by triality. If t is replaced by ts

with s ∈ O+ (n), the multiplier of t will not be changed and the multiplier of t+

will be multiplied by the multiplier µ(s+ ) of s+ . By Proposition (??) we have

µ(s+ )F —2 = Sn(s). Since n is multiplicative, Sn O+ (n) ≡ D(n) mod F —2 and

we can choose s (as a product of re¬‚ections) such that Sn(s) = »+ µ(t+ )’1 F —2 ,

hence the claim.

Using triality we de¬ne an action of A3 on PGO+ (n)(F ) = PGO+ (n): Let

[t] be the class of t ∈ GO+ (n)(F ) modulo the center. We put θ + ([t]) = [t+ ] and

θ’ ([t]) = [t’ ] where t+ , t’ are as in Proposition (??).

(35.6) Proposition. The maps θ + and θ’ are outer automorphisms of the group

PGO+ (n). They satisfy (θ + )3 = 1 and θ+ θ’ = 1, and they generate a subgroup of

Aut PGO+ (n) isomorphic to A3 .

Proof : It follows from the multiplicativity of the formulas in Proposition (??) that

the maps θ+ and θ’ are group homomorphisms. The relations among them also

follow from (??). Hence they are automorphisms and generate a homomorphic

image of A3 . The fact that θ + is not inner follows from Proposition (??).

We shall see that the action of A3 is, in fact, de¬ned on the group scheme

PGO+ (n) = GO+ (n)/ Gm . For this we need triality for Spin(n).

484 VIII. COMPOSITION AND TRIALITY

35.C. The group Spin and triality. The group scheme Spin(S, n) is de-

¬ned as

Spin(S, n)(R) = { c ∈ C0 (n)— | cSR c’1 ‚ SR and c„ (c) = 1 }

R

for all R ∈ Alg F . The isomorphism ±S of (??) can be used to give a nice description

of Spin(S, n) = Spin(S, n)(F ).

(35.7) Proposition. Assume that char F = 2. There is an isomorphism

{ (t, t+ , t’ ) | t, t+ , t’ ∈ O+ (S, n), t(x y) = t’ (x) t+ (y) }

Spin(S, n)

such that the vector representation χ : Spin(S, n) ’ O+ (S, n) corresponds to the

map (t, t+ , t’ ) ’ t. The other projections (t, t+ , t’ ) ’ t+ and (t, t+ , t’ ) ’ t’

correspond to the half-spin representations χpm of Spin(S, n).

+

Proof : Let c ∈ Spin(n) and let ±S (c) = t0 t0 . The condition cxc’1 = χc (x) ∈ S

’

for all x ∈ S is equivalent to the condition t (x y) = χc (x) t’ (y) or, by Propo-

+

sition (??), to χc (x y) = t’ (x) t+ (y) for all x, y ∈ S. Since by Proposition (??)

we have

ˆ’1 t+ —ˆ

b b 0

±S „ (c) = ˆ’1 t’ —ˆ

0 b b

where ˆ : S ’ S — is the adjoint of bn , the condition c„ (c) = 1 is equivalent to

∼

b ’

— —

t+ ˆ + = ˆ and t’ ˆ ’ = ˆ i.e., the t± are isometries of b = bn , hence of n since

bt b bt b,

char F = 2. They are proper by Proposition (??). Thus, putting

T (S, n) = { (t, t+ , t’ ) | t+ , t, t’ ∈ O+ (S, n), t(x y) = t’ (x) t+ (y) },

c ’ (χc , t+ , t’ ) de¬nes an injective group homomorphism φ : Spin(S, n) ’ T (S, n).

It is also surjective, since, given (t, t+ , t’ ) ∈ T (S, n), we have (t, t+ , t’ ) = φ(c) for

+

±S (c) = t0 t0 .’

From now on we assume that char F = 2. The isomorphism (??) can be de¬ned

on the level of group schemes: let G be the group scheme

t, t+ , t’ ∈ O+ (S, n)(R), t(x y) = t’ (x) t+ (y) .

(t, t+ , t’ )

G(R) =

∼

(35.8) Proposition. There exists an isomorphism β : G ’ Spin(S, n) of group

’

schemes.

Proof : By de¬nition we have

t+ 0

(±S — 1R )’1 = c ∈ Spin(S, n)(R),

0 t’

so that ±S induces a morphism

β : G ’ Spin(S, n).

Proposition (??) implies that β(Falg ) is an isomorphism. Thus, in view of Propo-

sition (??), it su¬ces to check that dβ is injective. It is easy to check that

Lie(G) = { (», »+ , »’ ) ∈ o(n) — o(n) — o(n) | »(x y) = »’ (x) y + x »+ (y) }.

On the other hand we have (see §??)

Lie Spin(S, n) = { x ∈ S · S ‚ C0 (S, n) | x + σ(x) = 0 } = [S, S]

where multiplication is in C0 (S, n) (recall that we are assuming that char F = 2

here) and the proof of Proposition (??) shows that that dβ is an isomorphism.

§35. CLIFFORD ALGEBRAS AND TRIALITY 485

Identifying Spin(S, n) with G through β we may de¬ne an action of A3 on

Spin(n):

(35.9) Proposition. The transformations θ + , resp. θ’ induced by

(t, t+ , t’ ) ’ (t+ , t’ , t), (t, t+ , t’ ) ’ (t’ , t, t+ )

resp.

3

are outer automorphisms of Spin(S, n) and satisfy the relations θ + = 1 and

’1

θ+ = θ’ . They generate a subgroup of Aut Spin(S, n) isomorphic to A3 . Fur-

thermore Spin(S, n)A3 is isomorphic to Autalg (C), if S is a para-Cayley algebra C,

and isomorphic to Autalg (A), resp. to Autalg (B, „ ), for a central simple algebra A

of degree 3, resp. a central simple algebra (B, „ ) of degree 3 with an involution of

second kind over K = F [X]/(X 2 + X + 1), if (S, ) is of type 1A2 , resp. of type 2A2 .

Proof : Let R be an F -algebra. It follows from the multiplicativity of the formulas

+ ’

of Proposition (??) that the maps θR and θR are automorphisms of Spin(S, n)(R).

The relations among them also follow from (??). They are outer automorphisms

since they permute the vector and the two half-spin representations of the group

Spin(S, n)(R) (since char F = 2, this also follows from the fact that they act non-

trivially on the center C, see Lemma (??)). Now let (t, t+ , t’ ) ∈ Spin(S, n)A3 (R).

We have t = t+ = t’ and t is an automorphism of SR . Conversely, any auto-

morphism of (S, n)R is an isometry and, since ±(x y) = ±(x) ±(y), ± is proper

by (??).

Let Spin8 = Spin(V, q) where V is 8-dimensional and q is hyperbolic.

(35.10) Corollary. (1) There exists an action of A3 on Spin8 such that SpinA3 8

is split of type G2 .

(2) There exists an action of A3 on Spin8 such that SpinA3 = PGU3 (K) where

8

2

K = F [X]/(X +X +1). In particular, if F contains a primitive cube root of unity,

there exists an action of A3 on Spin8 such that SpinA3 = PGL3 .

8

Proof : Take A = F — Cs , resp. A = M3 (K) in Proposition (??).

As we shall see in Proposition (??), the actions described in (??) and (??) of

Corollary (??) are (up to isomorphism) the only possible ones over Fsep .

Let again (S, n) be a symmetric composition of dimension 8 and norm n. In

view of (??) (and (??)) the group scheme Spin(S, n) ¬ts into the exact sequence

χ

1 ’ µ2 ’ Spin(S, n) ’ O+ (n) ’ 1

’

where χ is the vector representation, i.e.,

(χc )R (x) = cxc’1 for x ∈ SR , c ∈ Spin(S, n)(R).

Let χ : Spin(S, n) ’ PGO+ (n) be the composition of the vector representation χ

with the canonical map O+ (n) ’ PGO+ (n). The kernel C of χ is the center

of Spin(n) and is isomorphic to µ2 — µ2 . The action of A3 on Spin(S, n) restricts

to an action of A3 on µ2 — µ2 . We recall that we are still assuming char F = 2.

(35.11) Lemma. The action of A3 on C µ2 — µ2 is described by the exact

sequence

1 ’ C ’ µ2 — µ2 — µ2 ’ µ2 ’ 1

where A3 acts on µ2 —µ2 —µ2 through permutations and the map µ2 —µ2 —µ2 ’ µ2

is the multiplication map.

486 VIII. COMPOSITION AND TRIALITY

Proof : In fact we have

(35.12)

CR = {(1, 1, 1), = (1, ’1, ’1), = (’1, 1, ’1), = = (’1, ’1, 1)},

0 1 2 01

hence the description of C through the exact sequence. For the claim on the action

of A3 , note that θ+ = (θ’ )’1 maps i to i+1 with subscripts taken modulo 3.

Observe that the full group S3 acts on C and that Aut(C) = S3 .

In view of (??) (and (??)) we have an exact sequence

χ

1 ’ C ’ Spin(S, n) ’ PGO+ (n) ’ 1.

(35.13) ’

(35.14) Proposition. There is an outer action of A3 on PGO+ (n) such that the

maps in the exact sequence (??) above are A3 -equivariant.

Proof : The existence of the A3 action follows from Proposition (??), Lemma (??)

and the universal property (??) of factor group schemes. The action is outer, since

it is outer on Spin(S, n). Observe that for F -valued points the action is the one

de¬ned in Proposition (??).

Let (C, , n) be a para-Cayley algebra over F . The conjugation map π : x ’

Z/2Z

x can be used to extend the action of A3 to an action of S3 = A3

∼

+

on Spin(n) and PGO (n): Let ±C : C(C, n) ’ EndF (C • C), σn ⊥ n be the iso-

’

morphism of Proposition (??). The conjugation map x ’ x is an isometry and

since ±C C(π)±’1 = Int π π , π is improper. For any similitude t with multiplier

0

0

C

µ(t), t = µ(t) πtπ is a similitude with multiplier µ(t)’1 and is proper if and only

’1

if t is proper. Proposition (??) implies that

µ(t)’1 t(x y) = t+ (x) t’ (y)

(1)

µ(t’ )’1 t’ (x y) = t(x) t+ (y)

(2)

µ(t+ )’1 t+ (x y) = t’ (x) t(y)

(3)

hold in (C, ) if t is proper. Let θ + and θ’ be as de¬ned in (??) and (??). We

further de¬ne for R an F -algebra, θ([t]) = [t] for [t] ∈ PGO+ (n)(R), θ(t, t+ , t’ ) =

(t, t’ , t+ ) for (t, t+ , t’ ) ∈ G(R) Spin(n)(R) and θ( 0 ) = 0 , θ( 1 ) = 2 , θ( 2 ) = 1

for i as in (??).

(35.15) Proposition. (1) Let G be Spin(n) or PGO+ (n). The maps θ, θ +

and θ’ are outer automorphisms of G. They satisfy the relations

3 ’1

θ+ = 1, θ+ = θ’ θ+ θ = θθ’ ,

and

and they generate a subgroup of Aut(G) isomorphic to S3 . In particular Out(G)

S3 and Aut(G) PGO+ (n) S3 .

(2) The exact sequence

χ

1 ’ C ’ Spin(C, n) ’ PGO+ (C, n) ’ 1

’

is S3 -equivariant.

Proof : The proof is similar to the proof of Proposition (??) using the above formu-

las (??) to (??). In (??) the action of S3 on C is as de¬ned in (??). The fact that

S3 is the full group Out(G) follows from the fact that the group of automorphisms

of the Dynkin diagram of Spin(C, n), which is of type D4 , is S3 (see §??).

§35. CLIFFORD ALGEBRAS AND TRIALITY 487

Observe that for the given action of S3 on Spin(C, n) we have

Spin(C, n)S3 = Spin(C, n)A3 = Autalg (C).

(35.16)

The action of S3 on Spin8 induces an action on H 1 (F, Spin8 ). We now describe the

objects classi¬ed by H 1 (F, Spin8 ) and the action of S3 on H 1 (F, Spin8 ). A triple of

quadratic spaces (Vi , qi ), i = 1, 2, 3, together with a bilinear map β : V0 — V1 ’ V2

such that q2 β(v0 , v1 ) = q0 (v0 )q1 (v1 ) for vi ∈ Vi is a composition of quadratic

spaces. Examples are given by Vi = C, C a Cayley algebra, β(x, y) = x y, and by

∼

Vi = C, β(x, y) = x y. An isometry ψ : (V0 , V1 , V2 ) ’ (V0 , V1 , V2 ) is a triple of