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(2) (V, q) is similar to the norm n of a Cayley algebra C.
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

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