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11. Show that for n dividing 24, µn — µn and Z/nZ are isomorphic as Galois
modules.
12. Let (A, σ) be a central simple algebra over F with a symplectic involution σ.
Show that the map
Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, SL1 (A) = F — / Nrd(A— )
induced by the inclusion Sp(A, σ) ’ SL1 (A) takes a ∈ Sym(A, σ)— to its
pfa¬an norm NrpA (a) modulo Nrd(A— ).
13. Let A be a central simple algebra over F . For any c ∈ F — write Xc for the set
of all x ∈ A— such that Nrd(x) = c. Prove that
sep
(a) Xc is a SL1 (Asep )-torsor.
(b) Any SL1 (Asep )-torsor is isomorphic to Xc for some c.
(c) Xc Xd if and only if cd’1 ∈ Nrd(A— ).
14. Describe H 1 F, Spin(V, q) in terms of twisted forms of tensors.
15. Let (A, σ, f ) be a central simple F -algebra with quadratic pair of even degree 2n
over an arbitrary ¬eld F . Let Z be the center of the Cli¬ord algebra C(A, σ, f )
and let „¦(A, σ, f ) be the extended Cli¬ord group.
(a) Show that the connecting map
δ 1 : H 1 F, PGO+ (A, σ, f ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)
EXERCISES 445


in the cohomology sequence associated to
χ
1 ’ RZ/F (Gm,Z ) ’ „¦(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

’1
maps the 4-tuple (A , σ , f , •) to C(A , σ , f )—Z Z C(A, σ, f ) , where
the tensor product is taken with respect to •.
(b) Show that the multiplication homomorphism
Spin(A, σ, f ) — RZ/F (Gm,Z ) ’ „¦(A, σ, f )
induces an isomorphism
„¦(A, σ, f ) Spin(A, σ, f ) — RZ/F (Gm,Z ) /C
where C is isomorphic to the center of Spin(A, σ, f ). Similarly, show that
GO+ (A, σ, f ) O+ (A, σ, f ) — Gm /µ2
where µ2 is embedded diagonally in the product.
(c) Assume that n is even. Let ± : „¦(A, σ, f ) ’ GO+ (A, σ, f ) be the ho-
momorphism which, under the isomorphism in (??), is the vector rep-
resentation χ on Spin(A, σ, f ) and the norm map on RZ/F (Gm,Z ). By
relating via ± the exact sequence in (??) to a similar exact sequence for
GO+ (A, σ, f ), show that for all 4-tuple (A , σ , f , •) representing an ele-
ment of H 1 F, PGO+ (A, σ, f ) ,
’1
= [A ][A]’1
NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) in Br(F ).
In particular, NZ/F C(A, σ, f ) = [A].
Similarly, using the homomorphism „¦(A, σ, f ) ’ RZ/F (Gm,Z ) which is
trivial on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that
’1 2
C(A , σ , f ) —Z Z C(A, σ, f ) = 1.
2
In particular, C(A, σ, f ) = 1. (Compare with (??).)
(d) Assume that n is odd. Let G = O+ (A, σ, f ) — RZ/F (Gm,Z ) /µ2 . Using
the homomorphism ± : „¦(A, σ, f ) ’ G which is the vector representation
χ on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that for all
4-tuple (A , σ , f , •) representing an element of H 1 F, PGO+ (A, σ, f ) ,
’1 2
= [AZ ][AZ ]’1
C(A , σ , f ) —Z Z C(A, σ, f ) in Br(Z).
2
In particular, C(A, σ, f ) = [AZ ].
Using the character of „¦(A, σ, f ) which is trivial on Spin(A, σ, f ) and is
the norm on RZ/F (Gm,Z ), show that
’1
NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) = 1.
In particular, NZ/F C(A, σ, f ) = 1. (Compare with (??).)
16. (Qu´guiner [?]) Let (B, „ ) be a central simple F -algebra with unitary involution
e
of degree n. Let K be the center of B and let „ = Int(u) —¦ „ for some unit u ∈
Sym(B, „ ). Assume that char F does not divide n. Show that the Tits classes
t(B, „ ) and t(B, „ ) in H 2 (F, µn[K] ) are related by t(B, „ ) = t(B, „ ) + ζK ∪
NrdB (u) where ζK is the nontrivial element of H 1 (F, Z[K] ) and NrdB (u) =
NrdB (u) · F —n ∈ F — /F —n = H 1 (F, µn ). (Compare with (??).)
446 VII. GALOIS COHOMOLOGY


Notes
§??. The concept of a nonabelian cohomology set H 1 (“, A) has its origin in the
theory of principal homogeneous spaces (or torsors) due to Grothendieck [?], see
also Frenkel [?] and Serre [?]. The ¬rst steps in the theory of principal homogeneous
spaces attached to an algebraic group (in fact a commutative group variety) are
found in Weil [?].
Galois descent was implicitly used by Chˆtelet [?], in the case where A is an
a
elliptic curve (see also [?]). An explicit formulation (and proof) of Galois descent
in algebraic geometry was ¬rst given by Weil [?]. The idea of twisting the action
of the Galois group using automorphisms appears also in this paper, see Weil™s
commentaries in [?, pp. 543“544].
No Galois cohomology appears in the paper [?] on principal homogeneous spaces
mentioned above. The fact that Weil™s group of classes of principal homogeneous
spaces for a commutative group variety A over a ¬eld F stands in bijection with
the Galois 1-cohomology set H 1 (F, A) was noticed by Serre; details are given in
Lang and Tate [?], see also Tate™s Bourbaki talk [?].
The ¬rst systematic treatment of Galois descent, including nonabelian cases
(linear groups, in particular PGLn with application to the Brauer group), appeared
in Serre™s book “Corps locaux” [?], which was based on a course at the Coll`ge de
e
France in 1958/59. Twisted forms of algebraic structures viewed as tensors are
mentioned as examples. Applications to quadratic forms are given in Springer [?].
Another early application is the realization by Weil [?], following an observation of
“un amateur de cocycles tr`s connu”33 , of Siegel™s idea that classical groups can be
e
described as automorphism groups of algebras with involution (Weil [?, pp. 548“
549]).
Since then this simple but very useful formalism found many applications. See
the latest revised and completed edition of the Lecture Notes of Serre [?] and his
Bourbaki talk [?] for more information and numerous references. A far-reaching
generalization of nonabelian Galois cohomology, which goes beyond Galois exten-
sions and applies in the setting of schemes, was given by Grothendieck [?].
Our presentation in this section owes much to Serre™s Lecture Notes [?] and to
the paper [?] of Borel and Serre. The technique of changing base points by twisting
coe¬cients in cohomology, which we use systematically, was ¬rst developed there.
Note that the term “co-induced module” is used by Serre [?] and by Brown [?] for
the modules which we call “induced”, following Serre [?].
§??. Lemma (??), the so-called “Shapiro lemma”, was independently proved
by Eckmann [?, Theorem 4], D. K. Faddeev [?], and Arnold Shapiro. Shapiro™s
proof appears in Hochschild-Nakayama [?, Lemma 1.1].
Besides algebras and quadratic forms, Severi-Brauer varieties also have a nice
interpretation in terms of Galois cohomology: the group scheme PGL n occurs not
only as the automorphism group of a split central simple algebra of degree n, but
also as the automorphism group of the projective space Pn’1 . The Severi-Brauer
variety SB(A) attached to a central simple algebra A is a twisted form of the
projective space, given by the cocycle of A (see Artin [?]).
For any quadratic space (V, q) of even dimension 2n, the Cli¬ord functor de¬nes
a homomorphism C : PGO(V, q) ’ Autalg C0 (V, q) (see (??)). The induced map
in cohomology C 1 : H 1 F, PGO(V, q) ’ H 1 F, Autalg C0 (V, q) associates to
33 also referred to as “Mr. P. (the famous winner of many cocycle races)”
NOTES 447


every central simple F -algebra with quadratic pair of degree 2n a separable F -
algebra of dimension 22n’1 ; this is the de¬nition of the Cli¬ord algebra of a central
simple algebra with quadratic pair by Galois descent.
§??. Although the cyclic algebra construction is classical, the case considered
here, where L is an arbitrary Galois Z/nZ-algebra, is not so common in the lit-
erature. It can be found however in Albert [?, Chapter VII]. Note that if L is a
¬eld, its Galois Z/nZ-algebra structure designates a generator of the Galois group
Gal(L/F ).
The exact sequence (??) was observed by Arason-Elman [?, Appendix] and by
Serre [?, Chapter I, §2, Exercise 2]. (This exercise is not in the 1973 edition.) The
special case where M = µ2 (Fsep ) (Corollary (??)) plays a crucial rˆle in Arason [?].
o
The cohomological invariants f1 , g2 , f3 for central simple F -algebras with
unitary involution of degree 3 are discussed in Haile-Knus-Rost-Tignol [?, Corol-
lary 32]. It is also shown in [?] that these invariants are not independent and that
the invariant g2 (B, „ ) gives information on the ´tale F -subalgebras of B. To state
e
precise results, recall from (??) that cubic ´tale F -algebras with discriminant ∆
e
1
are classi¬ed by the orbit set H (F, A3[∆] )/S2 . Suppose char F = 2, 3 and let
F (ω) = F [X]/(X 2 + X + 1), so that µ3 = A3[F (ω)] . Let (B, „ ) be a central simple
F -algebra with unitary involution of degree 3 and let L be a cubic ´tale F -algebra
e
1
with discriminant ∆. Let K be the center of B and let cL ∈ H (F, A3[∆] ) be a co-
homology class representing L. The algebra B contains a subalgebra isomorphic to
L if and only if g2 (B, „ ) = cL ∪d for some d ∈ H 1 (F, A3[K—F (ω)—∆] ). (Compare with
Proposition (??).) If this condition holds, then B also contains an ´tale subalgebra
e
L with associated cohomology class d (hence with discriminant K — F (ω) — ∆).
Moreover, there exists an involution „ such that Sym(B, „ ) contains L and L .
See [?, Proposition 31].
§??. Let (A, σ) be a central simple algebra with orthogonal involution of even
degree over a ¬eld F of characteristic di¬erent from 2. The connecting homomor-
phism

δ 1 : H 1 F, O+ (A, σ) ’ H 2 (F, µ2 ) = 2 Br(F )

in the cohomology sequence associated to the exact sequence

1 ’ µ2 ’ Spin(A, σ) ’ O+ (A, σ) ’ 1

is described in Garibaldi-Tignol-Wadsworth [?]. Recall from (??) the bijection

H 1 F, O+ (A, σ) SSym(A, σ)— /≈.

For (s, z) ∈ SSym(A, σ)— , consider the algebra A = M2 (A) EndA (A2 ) with the
involution σ adjoint to the hermitian form 1, ’s’1 , i.e.,

’σ(c)s’1
a b σ(a)
σ = for a, b, c, d ∈ A.
sσ(d)s’1
c d ’sσ(b)

01
∈ A . We have s ∈ Skew(A , σ ) and NrdA (s ) = NrdA (s) = z 2 .
Let s =
s0
Therefore, letting Z be the center of the Cli¬ord algebra C(A , σ ) and

π : Skew(A , σ ) ’ Z
448 VII. GALOIS COHOMOLOGY


the generalized pfa¬an of (A , σ ) (see (??)), we have π(s )2 = z 2 . It follows that
1 ’1
2 1+z π(s ) is a nonzero central idempotent of C(A , σ ). Set
E(s, z) = 1 + z ’1 π(s ) · C(A , σ ),
a central simple F -algebra with involution of the ¬rst kind of degree 2deg A’1 . We
have
C(A , σ ) = E(s, z) — E(s, ’z)
and it is shown in Garibaldi-Tignol-Wadsworth [?, Proposition 4.6] that
δ 1 (s, z)/≈ = E(s, z) ∈ 2 Br(F ).
In particular, the images under δ 1 of (s, z) and (s, ’z) are the two components of
C(A , σ ). By (??), it follows that E(s, z) E(s, ’z) = [A], hence the Brauer class
E(s, z) is uniquely determined by s ∈ Sym(A, σ)— up to a factor [A]. This is the
invariant of hermitian forms de¬ned by Bartels [?]. Explicitly, let D be a division
F -algebra with involution of the ¬rst kind and let h be a nonsingular hermitian or
skew-hermitian form on a D-vector space V such that the adjoint involution σ = σh
on A = EndD (V ) is orthogonal. Let S = {1, [D]} ‚ Br(F ). To every nonsingular
form h on V of the same type and discriminant as h, Bartels attaches an invariant
c(h, h ) in the factor group Br(F )/S as follows: since h and h are nonsingular and
of the same type, there exists s ∈ Sym(A, σ)— such that
h (x, y) = h s’1 (x), y for all x, y ∈ V .
We have NrdA (s) ∈ F —2 since h and h have the same discriminant. We may then
set
c(h, h ) = E(s, z) + S = E(s, ’z) + S ∈ Br(F )/S
where z ∈ F — is such that z 2 = NrdA (s).
The Tits class t(B, „ ) ∈ H 2 (F, µn[K] ) for (B, „ ) a central simple F -algebra with
unitary involution of degree n with center K was de¬ned by Qu´guiner [?, §3.5.2], [?,
e
§2.2], who called it the determinant class. (Actually, Qu´guiner™s determinant class
e
di¬ers from the Tits class by a factor which depends only on n.)
All the material in §?? is based on unpublished notes of Rost (to appear). See
Serre™s Bourbaki talk [?].
Finally, we note that getting information for special ¬elds F on the set H 1 (F, G),
for G an algebraic group, gives rise to many important questions which are not ad-
dressed here. Suppose that G is semisimple and simply connected. If F is a p-adic
¬eld, then H 1 (F, G) is trivial, as was shown by Kneser [?]. If F is a number ¬eld,
the “Hasse principle” due to Kneser, Springer, Harder and Chernousov shows that
the natural map H 1 (F, G) ’ v H 1 (Fv , G) is injective, where v runs over the real
places of F and Fv is the completion of F at v. We refer to Platonov-Rapinchuk
[?, Chap. 6] for a general survey. If F is a perfect ¬eld of cohomological dimension
at most 2 and G is of classical type, Bayer-Fluckiger and Parimala [?] have shown
that H 1 (F, G) is trivial, proving Serre™s “Conjecture II” [?, Chap. III, §3] for clas-
sical groups. Analogues of the Hasse principle for ¬elds of virtual cohomological
dimension 1 or 2 were obtained by Ducros [?], Scheiderer [?] and Bayer-Fluckiger-
Parimala [?].
CHAPTER VIII


Composition and Triality

The main topic of this chapter is composition algebras. Of special interest from
the algebraic group point of view are symmetric compositions. In dimension 8 there
are two such types: Okubo algebras, related to algebras of degree 3 with unitary in-
volutions (type A2 ), and para-Cayley algebras related to Cayley algebras (type G 2 ).
The existence of these two types is due to the existence of inequivalent outer actions
of the group Z/3Z on split simply connected simple groups of type D4 (“triality”
for Spin8 ), for which the ¬xed elements de¬ne groups of type A2 , resp. G2 . Triality
is de¬ned here through an explicit computation of the Cli¬ord algebra of the norm
of an 8-dimensional symmetric composition. As a step towards exceptional simple
Jordan algebras, we introduce in the last section twisted compositions, generaliz-
ing a construction of Springer. The corresponding group of automorphisms is the
semidirect product Spin8 S3 .

§32. Nonassociative Algebras
In this and the following chapter, by an F -algebra A we mean (unless further
speci¬ed) a ¬nite dimensional vector space over F equipped with an F -bilinear mul-
tiplication m : A — A ’ A. We shall use di¬erent notations for the multiplication:
m(x, y) = xy = x y = x y. We do not assume in general that the multiplication
has an identity. An algebra with identity 1 is unital. An ideal of A is a subspace M
such that ma ∈ M and am ∈ M for all m ∈ M , a ∈ A. The algebra A is simple
if the multiplication on A is not trivial (i.e., there are elements a, b of A such that
ab = 0) and 0, A are the only ideals of A. The multiplication algebra M (A) is the
subalgebra of EndF (A) generated by left and right multiplications with elements
of A. The centroid Z(A) is the centralizer of M (A) in EndF (A):
Z(A) = { f ∈ EndF (A) | f (ab) = f (a)b = af (b) for a, b ∈ A }
and A is central if F ·1 = Z(A). If Z(A) is a ¬eld, the algebra A is central over Z(A).
Observe that a commutative algebra may be central if it is not associative.
The algebra A is strictly power-associative if, for every R ∈ Alg F , the R-
subalgebra of AR generated by one element is associative. We then write an for
nth -power of a ∈ A, independently of the notation used for the multiplication of
A. Examples are associative algebras, Lie algebras (trivially), alternative algebras,
i.e., such that
x(xy) = (xx)y and (yx)x = y(xx)
for all x, y ∈ A, and Jordan algebras in characteristic di¬erent from 2 (see Chap-
ter ??). Let A be strictly power-associative and unital. Fixing a basis (ui )1¤i¤r
of A and taking indeterminates {x1 , . . . , xr } we have a generic element
x= xi ui ∈ A — F (x1 , . . . , xr )
449
450 VIII. COMPOSITION AND TRIALITY


and there is a unique monic polynomial

PA,x (X) = X m ’ s1 (x)X m’1 + · · · + (’1)m sm (x) · 1

of least degree which has x as a root. This is the generic minimal polynomial of A.
The coe¬cients si are homogeneous polynomials in the xi ™s, s1 = TA is the generic
trace, sm = NA the generic norm and m is the degree of A. It is convenient to
view F as an algebra of degree n for any n such that char F does not divide n; the
corresponding polynomial is PF,x (X) = (X · 1 ’ x)n . In view of McCrimmon [?,
Theorem 4, p. 535] we have

NA (X · 1 ’ x) = PA,x (X)

for a strictly power-associative algebra A. For any element a ∈ A we can special-
ize the generic minimal polynomial PA,x (X) to a polynomial PA,a (X) ∈ F [X] by

writing a = i ai ui and substituting ai for xi . Let ± : A ’ A be an isomor-

phism of unital algebras. Uniqueness of the generic minimal polynomial implies
that PA ,±(x) = PA,x , in particular TA ±(x) = TA (x) and NA ±(x) = NA (x).

(32.1) Examples. (1) We have PA—B,(x,y) = PA,x · PB,y for a product algebra
A — B.
(2) For a central simple associative algebra A the generic minimal polynomial is
the reduced characteristic polynomial and for a commutative associative algebra it
is the characteristic polynomial.
(3) For a central simple algebra with involution we have a generic minimal poly-
nomial on the Jordan algebra of symmetric elements depending on the type of
involution:
An : If J = H(B, „ ), where (B, „ ) is central simple of degree n + 1 with a
unitary involution over a quadratic ´tale F -algebra K, PJ,a (X) is the restriction of
e
the reduced characteristic polynomial of B to H(B, „ ). The coe¬cients of PJ,a (X),
a priori in K, actually lie in F since they are invariant under ι. The degree of J is
the degree of B.
Bn and Dn : For J = H(A, σ), A central simple over F with an orthogonal
involution of degree 2n + 1, or 2n, PJ,a (X) is the reduced characteristic polynomial,
so that the degree of J is the degree of A.
Cn : For J = H(A, σ), A central simple of degree 2n over F with a symplec-
tic involution, PJ,a (X) is the polynomial Prpσ,a of (??). Here the degree of J
1
is 2 deg(A).

We now describe an invariance property of the coe¬cients si (x). Let s ∈ S(A— )
be a polynomial function on A, let d : A ’ A be an F -linear transformation, and
let F [µ] be the F -algebra of dual numbers. We say that s is Lie invariant under d
if

s a + µd(a) = s(a)

holds in A[µ] = A — F [µ] for all a ∈ A. The following result is due to Tits [?]:

(32.2) Proposition. The coe¬cients si (x) of the generic minimal polynomial of
a strictly power-associative F -algebra A are Lie invariant under all derivations d
of A.
§32. NONASSOCIATIVE ALGEBRAS 451


Proof : Let F be an arbitrary ¬eld extension of F . The extensions of the forms
si and d to AF will be denoted by the same symbols si and d. We de¬ne forms
{a, b}i and µi (a, b) by
(a + µb)i = ai + µ{a, b}i and si (a + µb) = si (a) + µµi (a, b).
It is easy to see (for example by induction) that d(ai ) = {a, d(a)}i for any deriva-
tion d. We obtain
0 = PA[µ],a+µb (a + µb)
n
n
(’1)i si (a) + µµi (a, b) an’i + µ{a, b}n’i ,
= a + µ{a, b}n +
i=1

where n is the degree of the generic minimal polynomial, so that
n n
i
(’1)i µi (a, b)an’i = 0.
(1) {a, b}n + (’1) si (a){a, b}n’i +
i=1 i=1

On the other hand we have
n
(’1)i si (a){a, d(a)}n’i = 0.
(2) d PA,a (a) = {a, d(a)}n +
i=1

Setting b = d(a) in (??) and subtracting (??) gives
n
(’1)i µi a, d(a) an’i = 0.
i=1

If a is generic over F , it does not satisfy any polynomial identity of degree n ’ 1.
Thus µi a, d(a) = 0. This is the Lie invariance of the si under the derivation d.

(32.3) Corollary. The identity s1 (a·b) = s1 (b·a) holds for any associative algebra
and the identity s1 a q (b q c) = s1 (a q b) q c holds for any Jordan algebra over a
¬eld of characteristic not 2.
Proof : The maps da (b) = a · b ’ b · a, resp. db,c (a) = a q (b q c) ’ (a q b) q c are
derivations of the corresponding algebras (see for example Schafer [?, p. 92] for the
last claim).

An algebra A is separable if A — F is a direct sum of simple ideals for every ¬eld
extension F of F . The following criterion (??) for separability is quite useful; it
applies to associative algebras and Jordan algebras in view of Corollary (??) and to
alternative algebras (see McCrimmon [?, Theorem 2.8]). For alternative algebras
of degree 2 and 3, which are the cases we shall consider, the lemma also follows
from (??) and Proposition (??). We ¬rst give a de¬nition: a symmetric bilinear
form T on an algebra A is called associative or invariant if
T (xy, z) = T (x, yz) for x, y, z ∈ A.
(32.4) Lemma (Dieudonn´). Let A be a strictly power-associative algebra with
e
generic trace TA . If the bilinear form T : (x, y) ’ TA (xy) is symmetric, nonsingular
and associative, then A is separable.
452 VIII. COMPOSITION AND TRIALITY


Proof : This is a special case of a theorem attributed to Dieudonn´, see for example
e
Schafer [?, p. 24]. Let I be an ideal. The orthogonal complement I ⊥ of I (with
respect to the bilinear form T ) is an ideal since T is associative. For x, y ∈ J = I©I ⊥
and z ∈ A, we have T (xy, z) = T (x, yz) = 0, hence J 2 = 0 and elements of J
are nilpotent. Nilpotent elements have generic trace 0 (see Jacobson [?, p. 226,
Cor. 1(2)]); thus T (x, z) = TA (xz) = 0 for all z ∈ A and x ∈ J. This implies J = 0
and A = I • I ⊥ . It then follows that A (and A — F for all ¬eld extensions F /F ) is
a direct sum of simple ideals, hence separable.

A converse of Lemma (??) also holds for associative algebras, alternative alge-
bras and Jordan algebras; a proof can be obtained by using Theorems (??) and (??).
Alternative algebras. The structure of ¬nite dimensional separable alterna-
tive algebras is similar to that of ¬nite dimensional separable associative algebras:

(32.5) Theorem. (1) Any separable alternative F -algebra is the product of simple
alternative algebras whose centers are separable ¬eld extensions of F .
(2) A central simple separable alternative algebra is either associative central simple
or is a Cayley algebra.

Reference: A reference for (??) is Schafer [?, p. 58]; (??) is a result due to Zorn,
see for example Schafer [?, p. 56]. We shall only use Theorem (??) for algebras of
degree 3. A description of Cayley algebras is given in the next section.

For nonassociative algebras the associator

(x, y, z) = (xy)z ’ x(yz)

is a useful notion. Alternative algebras are de¬ned by the identities

(x, x, y) = 0 = (x, y, y).

Linearizing we obtain

(32.6) (x, y, z) + (y, x, z) = 0 = (x, y, z) + (x, z, y),

i.e., in an alternative algebra the associator is an alternating function of the three
variables. The following result is essential for the study of alternative algebras:

(32.7) Theorem (E. Artin). Any subalgebra of an alternative algebra A generated
by two elements is associative.

Reference: See for example Schafer [?, p. 29] or Zorn [?].

Thus we have NA (xy) = NA (x)NA (y) and TA (xy) = TA (yx) for x, y ∈ A, A a
alternative algebra, since both are true for an associative algebra (see Jacobson [?,
Theorem 3, p. 235]). The symmetric bilinear form T (x, y) = TA (xy) is the bilinear
trace form of A.
In the next two sections separable alternative F -algebras of degree 2 and 3
are studied in detail. We set Sepalt n (m) for the groupoid of separable alternative
F -algebras of dimension n and degree m with isomorphisms as morphisms.
§33. COMPOSITION ALGEBRAS 453


§33. Composition Algebras
33.A. Multiplicative quadratic forms. Let C be an F -algebra with multi-
plication (x, y) ’ x y (but not necessarily with identity). We say that a quadratic
form q on C is multiplicative if
(33.1) q(x y) = q(x)q(y)
for all x, y ∈ C. Let bq (x, y) = q(x + y) ’ q(x) ’ q(y) be the polar of q and let
C ⊥ = { z ∈ C | bq (z, C) = 0 }.
(33.2) Proposition. The space C ⊥ is an ideal in C.
Proof : This is clear if q = 0. So let x ∈ C be such that q(x) = 0. Linearizing (??)
we have
bq (x y, x z) = q(x)bq (y, z).
Thus x y ∈ C ⊥ implies y ∈ C ⊥ . It follows that the kernel of the composed map
(of F -spaces)
p
φx : C ’x C ’ C/C ⊥ ,
’’
where x (y) = x y and p is the projection, is contained in C ⊥ . By dimension
count it must be equal to C ⊥ , so x C ⊥ ‚ C ⊥ and similarly C ⊥ x ‚ C ⊥ . Since
C ⊥ — L = (C — L)⊥ for any ¬eld extension L/F , the claim now follows from the
next lemma.
(33.3) Lemma. Let q : V ’ F be a nontrivial quadratic form. There exists a ¬eld
extension L/F such that V — L is generated as an L-linear space by anisotropic
vectors.
Proof : Let n = dimF V and let L = F (t1 , . . . , tn ). Taking n generic vectors in
V — L gives a set of anisotropic generators of V — L.
Let
R(C) = { z ∈ C ⊥ | q(z) = 0 }.
(33.4) Proposition. If (C, q) is a multiplicative quadratic form, then either C ⊥ =
R(C) or char F = 2 and C = C ⊥ .
Proof : We show that q|C ⊥ = 0 implies that char F = 2 and C = C ⊥ . If char F = 2,
then q(x) = 1 bq (x, x) = 0 for x ∈ C ⊥ , hence q|C ⊥ = 0 already implies char F =
2
2. To show that C = C ⊥ we may assume that F is algebraically closed. Since
char F = 2 the set R(C) is a linear subspace of C ⊥ ; by replacing C by C/R(C)
we may assume that R(C) = 0. Then q : C ⊥ ’ F is injective and semilinear with

respect to the isomorphism F ’ F , x ’ x2 . It follows that dimF C ⊥ = 1; let

u ∈ C ⊥ be a generator, so that q(u) = 0. For x ∈ C we have x u ∈ C ⊥ by
Proposition (??) and we de¬ne a linear form f : C ’ F by x u = f (x)u. Since
q(x)q(u) = q(x u) = q f (x)u = f (x)2 q(u)
q(x) = f (x)2 and the polar bq (x, y) is identically zero. This implies C = C ⊥ , hence
the claim.
(33.5) Example. Let (C, q) be multiplicative and regular of odd rank (de¬ned
on p. ??) over a ¬eld of characteristic 2. Since dimF C ⊥ = 1 and R(C) = 0,
Proposition (??) implies that C = C ⊥ and C is of dimension 1.
454 VIII. COMPOSITION AND TRIALITY


(33.6) Corollary. The set R(C) is always an ideal of C and q induces a multi-
plicative form q on C = C/R(C) such that either
(1) (C, q) is regular, or
(2) char F = 2 and C is a purely inseparable ¬eld extension of F of exponent 1 of
dimension 2n for some n and q(x) = x2 .
Proof : If R(C) = C ⊥ , R(C) is an ideal in C by Proposition (??) and the polar of q
is nonsingular. Then (??) follows from Corollary (??) except when dimF C = 1 in
characteristic 2. If R(C) = C ⊥ , then char F = 2 and C = C ⊥ by Proposition (??).
It follows that the polar bq (x, y) is identically zero, q : C ’ F is a homomorphism
and R(C) is again an ideal. For the description of the induced form q : C ’ F
we follow Kaplansky [?, p. 95]: the map q : C ’ F is an injective homomorphism,
thus C is a commutative associative integral domain. Moreover, for x such that
q(x) = 0, x2 /q(x) is an identity element 1 with q(1) = 1 and C is even a ¬eld. Since
q(» · 1) = »2 · 1 for all » ∈ F , we have
q(x2 ) = q q(x) · 1
(33.7)
for all x ∈ C. Let C0 = q(C), let x0 = q(x), and let be the induced multiplication.
0
It follows from (??) that
x0 x0 = q(x) · 1.
0

If dimF C = 1 we have the part of assertion (??) in characteristic 2 which was
left over. If dimF C > 1, then C is a purely inseparable ¬eld extension of F of
exponent 1, as claimed in (??).
(33.8) Remark. In case (??) of (??) C has dimension 1, 2, 4 or 8 in view of the
later Corollary (??).
33.B. Unital composition algebras. Let C be an F -algebra with identity
and multiplication (x, y) ’ x y and let n be a regular multiplicative quadratic form
on C. We call the triple (C, , n) a composition algebra. In view of Example (??),
the form 1 is the unique regular multiplicative quadratic form of odd dimension.
Thus it su¬ces to consider composition algebras with nonsingular forms in even
dimension ≥ 2. We then have the following equivalent properties:
(33.9) Proposition. Let (C, ) be a unital F -algebra with dimF C ≥ 2. The fol-
lowing properties are equivalent:
(1) There exists a nonsingular multiplicative quadratic form n on C.
(2) C is alternative separable of degree 2.
(3) C is alternative and has an involution π : x ’ x such that
x + x ∈ F · 1, n(x) = x x ∈ F · 1,
and n is a nonsingular quadratic form on C.
Moreover, the quadratic form n in (??) and the involution π in (??) are uniquely
determined by (C, ).
Proof : (??) ’ (??) Let (C, , n) be a composition algebra. To show that C is
alternative we reproduce the proof of van der Blij and Springer [?], which is valid
for any characteristic. Let
bn (x, y) = n(x + y) ’ n(x) ’ n(y)
§33. COMPOSITION ALGEBRAS 455


be the polar of n. The following formulas are deduced from n(x y) = n(x)n(y) by
linearization:
bn (x y, x z) = n(x)bn (y, z)
bn (x y, u y) = n(y)bn (x, u)
and
(33.10) bn (x z, u y) + bn (x y, u z) = bn (x, u)bn (y, z).
We have n(1) = 1. By putting z = x and y = 1 in (??), we obtain
bn x2 ’ bn (1, x)x + n(x) · 1, u = 0
for all u ∈ C. Since n is nonsingular any x ∈ C satisfy the quadratic equation
x2 ’ bn (1, x)x + n(x) · 1 = 0.
(33.11)
Hence C is of degree 2 and C is strictly power-associative. Furthermore b n (1, x)
is the trace TC (x) and n is the norm NC of C (as an algebra of degree 2). Let
x = TC (x) · 1 ’ x. It follows from (??) that
x x = x x = n(x) · 1
and it is straightforward to check that
x=x and 1 = 1.
Hence x ’ x is bijective. We claim that
(33.12) bn (x y, z) = bn (y, x z) = bn (x, z y).
The ¬rst formula follows from
bn (x y, z) + bn (y, x z) = bn (x, 1)bn (z, y) = TC (x)bn (z, y),
which is a special case of (??), and the proof of the second is similar. We further
need the formulas
x (x y) = n(x)y and (y x) x = n(x)y.
For the proof of the ¬rst one, we have
bn x (x y), z = bn (x y, x z) = bn n(x)y, z for z ∈ C.
The proof of the other one is similar. It follows that x (x y) = (x x) y.
Therefore
x (x y) = x TC (x)y ’ x y = TC (x)x ’ x x y = (x x) y
and similarly (y x) x = y (x x). This shows that C is an alternative algebra.
To check that the bilinear trace form T (x, y) = TC (x y) is nonsingular, we ¬rst
verify that π satis¬es π(x y) = π(y) π(x), so that π is an involution of C. By
linearizing the generic polynomial (??) we obtain
(33.13) x y + y x ’ TC (y)x + TC (x)y + bn (x, y)1 = 0.
On the other hand, putting u = z = 1 in (??) we obtain
bn (x, y) = TC (x)TC (y) ’ TC (x y)
(which shows that T (x, y) = TC (x y) is a symmetric bilinear form). By substituting
this in (??), we ¬nd that
TC (x) ’ x TC (y) ’ y = TC (y x) ’ y x,
456 VIII. COMPOSITION AND TRIALITY


thus π(x y) = π(y) π(x). It now follows that
TC (x y) = x y + x y = y x + x y = x y + y x = bn (x, y),
hence the bilinear form T is nonsingular if n is nonsingular. Furthermore TC (x y) =
bn (x, y) and (??) imply that
(33.14) T (x y, z) = T (x, y z) for x, y, z ∈ C,
hence, by Lemma (??), C is separable.
(??) ’ (??) Let
X 2 ’ TC (x)X + NC (x) · 1
be the generic minimal polynomial of C. We de¬ne π(x) = TC (x) ’ x and we put
n = NC ; then
x π(x) = π(x) x = n(x) · 1 ∈ F · 1
follows from x2 ’ x + π(x) x + n(x) · 1 = 0. Since bn (x, y) = T (x, y) and C is
separable, n is nonsingular. The fact that π is an involution follows as in the proof
of (??) ’ (??).
(??) ’ (??) The existence of an involution with the properties given in (??)
implies that C admits a generic minimal polynomial as given in (??). Since C is
alternative we have
x (x y) = n(x)y = (y x) x
Using that the associator (x, y, z) is an alternating function we obtain
n(x y) = (x y) (x y) = (x y) (y x)
= (x y) y x ’ (x y, y, x) = n(x)n(y) ’ (x, x y, y)
= n(x)n(y) ’ x (x y) y+x (x y) y
= n(x)n(y) ’ n(x)n(y) + n(x)n(y) = n(x)n(y)
so that n is multiplicative.
The fact that n and π are uniquely determined by (C, ) follows from the
uniqueness of the generic minimal polynomial.
Let Comp m be the groupoid of composition algebras of dimension m with iso-
morphisms as morphisms and let Comp + be the groupoid of unital composition
m
algebras with isomorphisms as morphisms.
(33.15) Corollary. The identity map C ’ C induces an isomorphism of groupoids
Comp+ ≡ Sepalt m (2) for m ≥ 2.
m

33.C. Hurwitz algebras. Let (B, π) be a unital F -algebra of dimension m
with an involution π such that
x + π(x) ∈ F · 1 and x π(x) = π(x) x ∈ F · 1
for all x ∈ B. Assume further that the quadratic form n(x) = x π(x) is nonsingular.
Let » ∈ F — . The Cayley-Dickson algebra CD(B, ») associated to (B, π) and » is
the vector space
CD(B, ») = B • vB
where v is a new symbol, endowed with the multiplication
(a + vb) (a + vb ) = a a + »b π(b) + v π(a) b + a b,
§33. COMPOSITION ALGEBRAS 457


for a, a , b and b ∈ A. In particular CD(B, ») contains B as a subalgebra and
v 2 = ».
Further we set
n(a + vb) = n(a) ’ »n(b) and π(a + vb) = π(a) ’ vb.
(33.16) Lemma. The algebra C = CD(B, ») is an algebra with identity 1 + v0
and π is an involution such that
TC (x) = x + π(x) ∈ F · 1, NC (x) = n(x) = x π(x) = π(x) x ∈ F · 1.
The algebra B is contained in CD(B, ») as a subalgebra and
(1) C is alternative if and only if B associative,
(2) C is associative if and only if B is commutative,
(3) C is commutative if and only if B = F .
Proof : The fact that C = CD(B, ») is an algebra follows immediately from the
de¬nition of C. Identifying v with v1 we have vB = v B and we view v as an
element of C. We leave the “if” directions as an exercise. The assertions about T C
and NC are easy to check, so that elements of C satisfy
x2 ’ TC (x)x + NC (x)1 = 0
and C is of degree 2. Thus, if C is alternative, n = NC is multiplicative by
Proposition (??). We have
n (a + v b) (c + v d) = n a c + »d b + v (c b + a d)
= n(a c + »d b) ’ »n(c b + a d),
on the other hand,
n (a + v b) (c + v d) = n(a + v b)n(c + v d)
= n(a) ’ »n(b) n(c) ’ »n(d) .
Comparing both expressions and using once more that n is multiplicative, we obtain
bn (a c, »d b) + n(v)bn (c b, a d) = 0
or, since n(v) = ’»,
bn (a c, d b) = bn (a d, c b)
so that
bn (a c) b, d = bn a (c b), d
for all a, b, c, d ∈ B by (??). Thus we obtain (a c) b = a (c b) and B is
associative. If C is associative, we have (v a) b = v (a b) = v(b a) and
b a = a b. Therefore B is commutative. Claim (??) is evident.
The passage from B to CD(B, ») is sometimes called a Cayley-Dickson process.
A quadratic ´tale algebra K satis¬es the conditions of Lemma (??) and the corre-
e
sponding Cayley-Dickson algebra Q = CD(K, ») is a quaternion algebra over F for
any » ∈ F — . Repeating the process leads to an alternative algebra CD(Q, µ). A
Cayley algebra is a unital F -algebra isomorphic to an algebra of the type CD(Q, µ)
for some quaternion algebra Q over F and some µ ∈ F — .
In view of Lemma (??) and Proposition (??), the Cayley-Dickson process ap-
plied to a Cayley algebra does not yield a composition algebra. We now come to
the well-known classi¬cation of unital composition algebras:
458 VIII. COMPOSITION AND TRIALITY


(33.17) Theorem. Composition algebras with identity element over F are ei-
ther F , quadratic ´tale F -algebras, quaternion algebras over F , or Cayley algebras
e
over F .
Proof : As already observed, all algebras in the list are unital composition algebras.
Conversely, let C be a composition algebra with identity element over F . If C = F ,
let c ∈ C be such that {1, c} generates a nonsingular quadratic subspace of (C, n):
choose c ∈ 1⊥ if char F = 2 and c such that bn (1, c) = 1 if char F = 2. Then
B = F · 1 • F · c is a quadratic ´tale subalgebra of C. Thus we may assume that C
e
contains a unital composition algebra with nonsingular norm and it su¬ces to show
that if B = C, then C contains a Cayley-Dickson process B+vB. If B = C, we have
C = B • B ⊥ , B ⊥ is nonsingular and there exists v ∈ B ⊥ such that n(v) = ’» = 0.
We claim that B • v B is a subalgebra of C obtained by a Cayley-Dickson process,
i.e., that
(v a) b = v (b a), a (v b) = v (a b)
and
(v a) (v b) = »b a
for a, b ∈ B. We only check the ¬rst formula. The proofs of the others are similar.
We have v = ’v, since bn (v, 1) = 0, and 0 = bn (v, a)·1 = v a+a v = ’v a+a v,
thus v a = a v = ’a v for a ∈ B. Further
bn (v b) a, z = bn (v b, z a) = bn (b v, z a) = ’bn (b a, z v).
The last equality follows from formula (??), putting x = b, u = z, y = a, z = v,
and using that bn (v, a) = 0 for a ∈ B. On the other hand we have
’bn (b a, z v) = ’bn (b a) v, z = bn v (a b), z
where the last equality follows from the fact that v a = ’a v for all a ∈ B. This
holds for all z ∈ C, hence (v b) a = v (a b) as claimed. The formulas for the
norm and the involution are easy and we do not check them.
The classi¬cation of composition algebras with identity is known as the The-
orem of Hurwitz and the algebras occurring in Theorem (??) are called Hurwitz
algebras.
From now on we set Comp + = Hurw m for m = 1, 2, 4, and 8. If Sm , A1 ,
m
resp. G2 , are the groupoids of ´tale algebras of dimension m, quaternion algebras,
e
resp. Cayley algebras over F , then Hurw 2 = S2 , Hurw 4 = A1 , and Hurw 8 = G2 .
Hurwitz algebras are related to P¬ster forms. Let PQ m be the groupoid of
P¬ster quadratic forms of dimension m with isometries as morphisms.
(33.18) Proposition. (1) Norms of Hurwitz algebras are 0-, 1-, 2-, or 3-fold
P¬ster quadratic forms and conversely, all 0-, 1-, 2- or 3-fold P¬ster quadratic
forms occur as norms of Hurwitz algebras.
(2) For any Hurwitz algebra (C, NC ) the space
(C, NC )0 = { x ∈ C | TC (x) = 0 },
where TC is the trace, is regular.
Proof : (??) This is clear for quadratic ´tale algebras. The higher cases follow from
e
the Cayley-Dickson construction.
§33. COMPOSITION ALGEBRAS 459


Similarly, (??) is true for quadratic ´tale algebras, hence for Hurwitz algebras
e
of higher dimension by the Cayley-Dickson construction.
(33.19) Theorem. Let C, C be Hurwitz algebras. The following claims are equiv-
alent:
(1) The algebras C and C are isomorphic.
(2) The norms NC and NC are isometric.
(3) The norms NC and NC are similar.
Proof : (??) ’ (??) follows from the uniqueness of the generic minimal polynomial

and (??) ’ (??) is obvious. Let now ± : (C, NC ) ’ (C , NC ) be a similitude with

factor ». Since NC ±(1C ) = »N (1C ) = », » is represented by NC . Since NC is
a P¬ster quadratic form, »NC is isometric to NC (Baeza [?, p. 95, Theorem 2.4]).
Thus we may assume that ± is an isometry. Let dimF C ≥ 2 and let B1 be a
quadratic ´tale subalgebra of C such that its norm is of the form [1, b] = X 2 +
e
2
XY +bY with respect to the basis (1, u) for some u ∈ B1 . Let ±(1) = e, ±(u) = w,
and let e be the left multiplication with e. Then u = e (w) generates a quadratic

´tale subalgebra B1 of C and β = e —¦ ± is an isometry NC ’ NC which restricts
e ’

to an isomorphism B1 ’ B1 . Thus we may assume that the isometry ± restricts

to an isomorphism on a pair of quadratic ´tale algebras B1 and B1 . Then ± is
e

∼ ⊥∼
an isometry NB1 ’ NB1 , hence induces an isometry B1 ’ B1 . If B1 = C,
’ ’

choose v ∈ B1 such that N (v) = 0 and put v = ±(v). By the Cayley-Dickson
construction (??) we may de¬ne an isomorphism

±0 : B 2 = B 1 • v B 1 ’ B 2 = B 1 • v
’ B1
by putting ±0 (a + v b) = ±(a) + v ±(b) (which is not necessarily equal to
±(a + v b)!). Assume that B2 = C. Since ±0 is an isometry, it can be extended by

Witt™s Theorem to an isometry C ’ C . We now conclude by repeating the last

step.
(33.20) Corollary. There is a natural bijection between the isomorphism classes
of Hurw m and the isomorphism classes of PQ m for m = 1, 2, 4, and 8.
Proof : By (??) and (??).
The following “Skolem-Noether” type of result is an immediate consequence of
the proof of the implication (??) ’ (??) of (??):
(33.21) Corollary. Let C1 , C2 be separable subalgebras of a Hurwitz algebra C.

Any isomorphism φ : C1 ’ C2 extends to an isomorphism or an anti-isomorphism

of C.
(33.22) Remark. It follows from the proof of Theorem (??) that an isometry
of a quadratic or quaternion algebra which maps 1 to 1 is an isomorphism or an
anti-isomorphism (“A1 ≡ B1 ”). This is not true for Cayley algebras (“G2 ≡ B3 ”).
(33.23) Proposition. If the norm of a Hurwitz algebra is isotropic, it is hyper-
bolic.
Proof : This is true in general for P¬ster quadratic forms (Baeza [?, Corollary 3.2,
p. 105]), but we still give a proof, since it is an easy consequence of the Cayley-
Dickson process. We may assume that dimF C ≥ 2. If the norm of a Hurwitz
algebra C is isotropic, it contains a hyperbolic plane and we may assume that 1C
460 VIII. COMPOSITION AND TRIALITY


lies in this plane. Hence C contains the split separable F -algebra B = F — F . But
then any B • vB obtained by the Cayley-Dickson process is a quaternion algebra
with zero divisors, hence a matrix algebra, and its norm is hyperbolic. Applying
once more the Cayley-Dickson process if necessary shows that the norm must be
hyperbolic if dimF C = 8.
It follows from Theorem (??) and Proposition (??) that in each possible dimen-
sion there is only one isomorphism class of Hurwitz algebras with isotropic norms.
For Cayley algebras a model is the Cayley algebra
Cs = CD M2 (F ), ’1 .
We call it the split Cayley algebra. Its norm is the hyperbolic space of dimen-
sion 8. The group of F -automorphisms of the split Cayley algebra Cs over F is an
exceptional simple split group G of type G2 (see Theorem (??)).
(33.24) Proposition. Let G be a simple split algebraic group of type G 2 . Cayley
algebras over a ¬eld F are classi¬ed by the pointed set H 1 (F, G).
Proof : Since all Cayley algebras over a separable closure Fs of F are split, any
Cayley algebra over F is a form of the split algebra Cs . Thus we are in the situation
of (??), hence the claim.
If the characteristic of F is di¬erent from 2, norms of Hurwitz algebras corre-
spond to n-fold (bilinear) P¬ster forms for n = 0, 1, 2, and 3. We recall that for
any n-fold P¬ster form qn = a1 , . . . , an the element fn (qn ) = (±1 ) ∪ · · · ∪ (±n ) ∈
H n (F, µ2 ) is an invariant of the isometry class of qn and classi¬es the form. (see
Theorem (??)). Thus in characteristic not 2, the cohomological invariant fi (NC )
of the norm NC of a Hurwitz algebra C of dimension 2i ≥ 2 is an invariant of the
algebra. We denote it by fi (C) ∈ H i (F ).
(33.25) Corollary. Let C, C be Hurwitz algebras of dimension 2i , i ≥ 1. The
following conditions are equivalent:
(1) The algebras C and C are isomorphic.
(2) fi (C) = fi (C ).
Proof : By Theorem (??) and Theorem (??).
(33.26) Remark. There is also a cohomological invariant for P¬ster quadratic
forms in characteristic 2 (see for example Serre [?]). For this invariant, Theo-
rem (??) holds, hence, accordingly, Corollary (??) also.
33.D. Composition algebras without identity. We recall here some gen-
eral facts about composition algebras without identity, as well as consequences of
previous results for such algebras.
The norm n of a composition algebra is determined by the multiplication even
if C does not have an identity:
(33.27) Proposition. (1) Let (C, , n) be a composition algebra with multiplica-
tion (x, y) ’ x y, not necessarily with identity. Then there exists a multiplication
(x, y) ’ x y on C such that (C, , n) is a unital composition algebra with respect
to the new multiplication.
(2) Let (C, , n), (C , , n ) be composition algebras (not necessarily with identity).
∼ ∼
Any isomorphism of algebras ± : (C, ) ’ (C , ) is an isometry (C, n) ’ (C , n ).
’ ’
§34. SYMMETRIC COMPOSITIONS 461


Proof : (??) (Kaplansky [?]) Let a ∈ C be such that n(a) = 0 and let u = n(a)’1 a2 ,
so that n(u) = 1. The linear maps u : x ’ u x and ru : x ’ x u are isometries,
hence bijective. We claim that v = u2 is an identity for the multiplication
’1 ’1
x y = (ru x) ( u y).
We have ru v = ’1 v = u, hence x
’1 ’1 ’1 ’1
v= (ru x) ( u v) = ru (ru x) = x and
u
similarly v x = x. Furthermore,
’1 ’1 ’1 ’1
n(x y) = n (ru x) ( u y) = n(ru x)n( u y) = n(x)n(y).
(??) (Petersson [?]) The claim follows from the uniqueness of the degree two
generic minimal polynomial if ± is an isomorphism of unital algebras. In particular
n is uniquely determined by if there is a multiplication with identity. Assume now
that C and C do not have identity elements. We use ± to transport n to C, so that
we have one multiplication on C which admits two multiplicative norms n, n .
If there exists some a ∈ C with n(a) = n (a) = 1, we modify the multiplication
as in (??) to obtain a multiplication with 1 which admits n and n , so n = n .
To ¬nd a, let u ∈ C be such that n(u) = 1 (such an element exists by (??)). The
map u : x ’ u x is then an isometry of (C, n), and in particular it is bijective.
This implies that n (u) = 0. The element a such u a = u has the desired property
n(a) = n (a) = 1.
(33.28) Corollary. The possible dimensions for composition algebras (not neces-
sarily unital ) are 1, 2, 4 or 8.
Proof : The claim follows from Theorem (??) for unital algebras and hence from
Proposition (??) in general.
(33.29) Corollary. Associating a unital composition algebra (C, , n) to a compo-
sition algebra (C, , n) de¬nes a natural map to the isomorphism classes of Hurw m
from the isomorphism classes of Comp m .
Proof : By Proposition (??) we have a unital multiplication on (C, n) which, by
Theorem (??), is determined up to isomorphism.
(33.30) Remark. As observed in Remark (??) an isometry of unital composition
algebras which maps 1 to 1 is not necessarily an isomorphism, however isometric
unital composition algebras are isomorphic. This is not necessarily the case for
algebras without identity (see Remark (??)).

§34. Symmetric Compositions
In this section we discuss a special class of composition algebras without iden-
tity, independently considered by Petersson [?], Okubo [?], Faulkner [?] and re-
cently by Elduque-Myung [?], Elduque-P´rez [?]. Let (S, n) be a ¬nite dimensional
e
F -algebra with a quadratic form n : S ’ F . Let bn (x, y) = n(x+y)’n(x)’n(y) for
x, y ∈ S and let (x, y) ’ x y be the multiplication of S. We recall that the quad-
ratic form n is called associative or invariant with respect to the multiplication
if
bn (x y, z) = bn (x, y z)
holds for all x, y, z ∈ S.
(34.1) Lemma. Assume that n is nonsingular. The following conditions are equiv-
alent:
462 VIII. COMPOSITION AND TRIALITY


(1) n is multiplicative and associative.
(2) n satis¬es the relations x (y x) = n(x)y = (x y) x for x, y ∈ S.

Proof : (Okubo-Osborn [?, Lemma II.2.3]) Assume (??). Linearizing n(x y) =
n(x)n(y), we obtain

bn (x y, x z) = bn (y x, z x) = n(x)bn (y, z).

Since n is associative, this yields

0 = bn (x y) x ’ n(x)y, z = bn y, x (z x) ’ n(x)z ,

hence (??), n being nonsingular.
Conversely, if (??) holds, linearizing gives

(34.2) x (y z) + z (y x) = bn (x, z)y = (x y) z + (z y) x.

By substituting x y for x in the ¬rst equation and y z for z in the second equation,
we obtain

(x y) (y z) = bn (x y, z)y ’ z (y x y) = bn (x, y z)y ’ (y z y) x
= bn (x y, z)y ’ n(y)z x = bn (x, y z)y ’ n(y)z x,

hence bn (x y, z)y = bn (x, y z)y and bn (x, y) is associative. Finally, we have

n(x y)y = (x y) [y (x y)] = (x y) n(y)x = n(y)n(x)y

and the form n is multiplicative. If 2 = 0, we can also argue as follows:

n(x y) = 2 bn (x y, x y) = 1 bn x, y (x y) = 2 bn (x, x)n(y) = n(x)n(y).
1 1
2




We call a composition algebra with an associative norm a symmetric compo-
sition algebra and denote the full subcategory of Comp m consisting of symmetric
composition algebras by Scomp m . A symmetric composition algebra is cubic, be-
cause

x (x x) = (x x) x = n(x)x,

however it is not power-associative in general, since

(34.3) (x x) (x x) = bn (x, x x)x ’ x x (x x)

by (??) and

x x (x x) = n(x)x x.

A complete list of power-associative symmetric composition algebras is given in Ex-
ercise ?? of this chapter.
The ¬eld F is a symmetric composition algebra with identity. However it can
be shown that a symmetric composition algebra of dimension ≥ 2 never admits an
identity.
§34. SYMMETRIC COMPOSITIONS 463


34.A. Para-Hurwitz algebras. Let (C, , n) be a Hurwitz algebra. The
multiplication
(x, y) ’ x y = x y
also permits composition and it follows from bn (x y, z) = bn (x, z y) (see (??)) that
the norm n is associative with respect to (but not with respect to if C = F ).
Thus (C, , n) is a symmetric composition algebra. We say that (C, , n) is the
para-Hurwitz algebra associated with (C, , n) (resp. the para-quadratic algebra, the
para-quaternion algebra or the para-Cayley algebra). We denote the corresponding
full subcategories of Scomp by Hurw , resp. S2 , A1 , and G2 .
Observe that the unital composition algebra associated with (C, ) by the con-
struction given in the proof of Proposition (??) is the Hurwitz algebra (C, ) if we
set a = 1.
(34.4) Proposition. Let (C1 , , n1 ) and (C2 , , n2 ) be Hurwitz algebras and let

± : C1 ’ C 2

be an isomorphism of vector spaces such that ±(1C1 ) = 1C2 . Then ± is an isomor-
∼ ∼
phism (C1 , ) ’ (C2 , ) of algebras if and only if it is an isomorphism (C1 , ) ’
’ ’
(C2 , ) of para-Hurwitz algebras. Moreover

(1) Any isomorphism of algebras (C1 , ) ’ (C2 , ) is an isomorphism of the cor-

responding para-Hurwitz algebras.

(2) If dim C1 ≥ 4, then an isomorphism (C1 , ) ’ (C2 , ) of para-Hurwitz algebras

is an isomorphism of the corresponding Hurwitz algebras.
Proof : Let ± : C1 ’ C2 be an isomorphism of algebras. By uniqueness of the
quadratic generic polynomial we have ±(x) = ±(x) and ± is an isomorphism of
para-Hurwitz algebras. Conversely, an isomorphism of para-Hurwitz algebras is an
isometry by Proposition (??) (or since x (y x) = n(x)y), and we have TC2 ±(x) =
TC1 (x), since TC1 (x) = bC1 (1, x) and ±(1C1 ) = 1C2 . As above it follows that
±(x) = ±(x) and ± is an isomorphism of Hurwitz algebras.
Claim (??) obviously follows from the ¬rst part and claim (??) will also follow
from the ¬rst part if we show that ±(1C1 ) = 1C2 . We use Okubo-Osborn [?, p. 1238]:
we have 1 x = ’x for x ∈ 1⊥ and the claim follows if we show that there exists
exactly one element u ∈ C1 such that u x = ’x for x ∈ u⊥ . Let u be such an
element. Since by Corollary (??.??), 1⊥ is nondegenerate, the maximal dimension
1
of a subspace of 1⊥ on which the form is trivial is 2 (dimF C1 ’ 2). If dimF C1 ≥ 4,
there exists some x ∈ 1⊥ © u⊥ with n1 (x) = 0. For this element x we have
n1 (x)1 = x (1 x) = x (’x) = x (u x) = n1 (x)u,
so that, as claimed, 1 = u.
For quadratic algebras the following nice result holds:
(34.5) Proposition. Let C1 , C2 be quadratic algebras and assume that there exists

an isomorphism of para-quadratic algebras ± : (C1 , ) ’ (C2 , ), which is not an

isomorphism of algebras. Then u = ±(1) ∈ F · 1 is such that u3 = 1 and β(x) =
∼ ∼
±(x)u2 is an algebra isomorphism C1 ’ C2 . Conversely, if β : C1 ’ C2 is an
’ ’
isomorphism of algebras, then, for any u ∈ C2 such that u3 = 1, the map ± de¬ned

by ±(x) = β(x)u is an isomorphism (C1 , ) ’ (C2 , ) of para-quadratic algebras.


In particular an isomorphism C1 ’ C2 of para-Hurwitz algebras which is not an

464 VIII. COMPOSITION AND TRIALITY


F [X]/(X 2 + X + 1), i.e., C1 is
isomorphism of algebras can only occur if C1
isomorphic to F (u) with u3 = 1.
Proof : The proof of (??) shows that u = ±(1) ∈ F . We show that u3 = 1. We have
u u = u2 = u. Thus multiplying by u and conjugating gives u3 = uu = n2 (u) =
n2 ±(1) = n1 (1) = 1 by Proposition (??). It then follows that C2 F (u). The
condition ±(x) ±(y) = ±(x y) with y = 1 gives ±(x)u = ±(x) and, replacing x
by xy,
±(xy)u = ±(xy) = ±(x)±(y).
By conjugating and multiplying both sides with u4 = u we obtain
[±(x)u2 ][±(y)u2 ] = ±(xy)u2 ,
so that the map β : C1 ’ C2 de¬ned by β(x) = ±(x)u2 is an isomorphism of

C2 = F (u) with u3 = 1 and if β : C1 ’ C2 is
algebras. Conversely, if C1 ’
an isomorphism, then ± : C1 ’ C2 de¬ned by ±(x) = β(x)u is an isomorphism

(C1 , ) ’ (C2 , ).

Observe that ru : x ’ x u is an automorphism of F (u), of order 3. In
fact we have AutF F (u), = S3 , generated by the conjugation and ru . This is in
contrast with the quadratic algebra F (u), · for which AutF F (u) = Z/2Z.
(34.6) Corollary. The map P : (C, ) ’ (C, ) is an equivalence Hurw m ≡ Hurw m
of groupoids if m = 4, 8, and P is bijective on isomorphism classes if m = 2.
In view of Corollary (??) we call a n-dimensional para-Hurwitz composition
algebra of type A1 if n = 4 and of type G2 if n = 8.
(34.7) Remark. It follows from Corollary (??) that
AutF (C, ) = AutF (C, )
for any Hurwitz algebra C of dimension ≥ 4. Thus the classi¬cation of twisted
forms of para-Hurwitz algebras is equivalent to the classi¬cation of Hurwitz alge-
bras in dimensions ≥ 4. In particular any twisted form of a para-Hurwitz algebra of
dimension ≥ 4 is again a para-Hurwitz algebra. The situation is di¬erent in dimen-
sion 2: There exist forms of para-quadratic algebras which are not para-quadratic
algebras (see Theorem (??)).
The identity 1 of a Hurwitz algebra C plays a special role also for the associated
para-Hurwitz algebra: it is an idempotent and satis¬es 1 x = x 1 = ’x for all
x ∈ C such that bnC (x, 1) = 0. Let (S, , n) be a symmetric composition algebra.
An idempotent e of S (i.e., an element such that e e = e) is called a para-unit if
e x = x e = ’x for x ∈ S, bn (e, x) = 0.
(34.8) Lemma. A symmetric composition algebra is para-Hurwitz if and only if
it admits a para-unit.
Proof : If (S, ) is para-Hurwitz, then 1 ∈ S is a para-unit. Conversely, for any
para-unit e in a symmetric composition algebra (S, , n), we have n(e) = 1 and
x y = (e x) (y e)
de¬nes a multiplication with identity element e on S. We have x y = x y where
x = bn (e, x)e ’ x.
§34. SYMMETRIC COMPOSITIONS 465


34.B. Petersson algebras. Let (C, , n) be a Hurwitz algebra and let • be
an F -automorphism of C such that •3 = 1. Following Petersson [?] we de¬ne a
new multiplication on C by
x y = •(x) •2 (y).
This algebra, denoted C• , is a composition algebra for the same norm n and we
call it a Petersson algebra. It is straightforward to check that
(x y) x = n(x)y = x (y x)
so that Petersson algebras are symmetric composition algebras. Observe that • is
automatically an automorphism of (C, ). For • = 1, (C, ) is para-Hurwitz.
Conversely, symmetric composition algebras with nontrivial idempotents are
Petersson algebras (Petersson [?, Satz 2.1], or Elduque-P´rez [?, Theorem 2.5]):
e
(34.9) Proposition. Let (S, , n) be a symmetric composition algebra and let e ∈
S be a nontrivial idempotent.
(1) The product x y = (e x) (y e) gives S the structure of a Hurwitz algebra
with identity e, norm n, and conjugation x ’ x = bn (x, e)e ’ x.
(2) The map
•(x) = e (e x) = bn (e, x)e ’ x e = x e
is an automorphism of (S, ) (and (S, )) of order ¤ 3 and (S, ) = S• is a Petersson
algebra with respect to •.
Proof : (??) is easy and left as an exercise.
(??) Replacing x by e x and z by e in the identity (??):
bn (x, z)y = x (y z) + z (y x)
gives
x y = bn (e, x)y ’ e y (e x)
hence
(x y) e = y bn (x, e)e ’ e x = e (y e) (x e) e = (y e) (x e).
Thus • is an automorphism of (S, ), •3 (x) = x = x, x •2 (y) and
y = •(x)
(S, ) = S• as claimed.

In general a symmetric composition may not contain an idempotent. However:
(34.10) Lemma. Let (S, , n) be a symmetric composition algebra.
(1) If the cubic form bn (x x, x) is isotropic on S, then (S, ) contains an idempo-
tent. In particular there always exists a ¬eld extension L/F of degree 3 such that
(S, )L contains an idempotent e.
(2) For any nontrivial idempotent e ∈ S we have n(e) = 1.
Proof : (??) It su¬ces to ¬nd f = 0 with f f = »f , » ∈ F — so that e = f »’1
then is an idempotent. Let x = 0 be such that bn (x x, x) = 0. We have
(x x) (x x) = ’n(x)(x x)
by (??), so we take f = x x if n(x) = 0. If n(x) = 0, we may also assume that
x x = 0: if x x = 0 we replace x by x x and use again (??). Since x is isotropic
466 VIII. COMPOSITION AND TRIALITY


and n is nonsingular, there exists some y ∈ S such that n(y) = 0 and bn (x, y) = ’1.
A straightforward computation using (??) shows that
(x y + y x) (x y + y x) = (x y + y x) + 3bn (y, y x)x,
and
e = x y + y x + bn (y, y x)x
is an idempotent and is nonzero since
e x = (y x) x = bn (x, y)x = ’x.
(??) Since e = (e e) e = n(e)e, we have n(e) = 1.

(34.11) Remark. Lemma (??.??) is in fact a special case of Theorem (??) and
its proof is copied from the proof of implication (??) ’ (??) of (??).
Assume that char F = 3 and that F contains a primitive cube root of unity ω.
The existence of an automorphism of order 3 on a Hurwitz algebra C is equivalent
with the existence of a Z/3Z-grading:
(34.12) Lemma. Suppose that F contains a primitive cube root of unity ω.
(1) If • is an automorphism of C of order 3, then C (or C• ) admits a decomposition
C = C • = S0 • S1 • S2 ,
with
Si = { x ∈ C | •(x) = ω i x }
and such that
(a) Si Sj ‚ Si+j (resp. Si Sj ‚ Si+j ), with subscripts taken modulo 3,
(b) bn (Si , Sj ) = 0 unless i + j ≡ 0 mod 3.
In particular (S0 , , n) ‚ C• is a para-Hurwitz algebra of even dimension and S1
(resp. S2 ) is a maximal isotropic subspace of S1 • S2 .
(2) Conversely, any Z/3Z-grading of C de¬nes an automorphism • of order 3 of C,
hence a Petersson algebra C• .
Proof : Claim (??) follows easily from the fact that ω i , i = 0, 1, 2, are the eigen-
values of the automorphism •. For (??) we take the identity on degree 0 elements,
multiplication by ω on degree 1 elements and multiplication by ω 2 on degree 2
elements.

If • = 1, S0 in Lemma (??) must have dimension 2 or 4 (being a para-Hurwitz
algebra). We show next that C• is para-Hurwitz if dim S0 = 2. The case dim S0 = 4
and dim C• = 8 corresponds to a di¬erent type of symmetric composition, discussed
in the next subsection.
(34.13) Proposition (Elduque-P´rez). Let F be a ¬eld of characteristic not 3, let
e
C be a Hurwitz algebra over F , let • be an F -automorphism of C of order 3. Then
S0 = { x ∈ C | •(x) = x }
is a para-Hurwitz algebra of dimension 2 or 4. The Petersson algebra C • is iso-
morphic to a para-Hurwitz algebra if and only if dim S0 = 2.
§34. SYMMETRIC COMPOSITIONS 467


Proof : The ¬rst claim is clear. For the second claim we use an argument in Elduque-
P´rez [?, proof of Proposition 3.4]. If dim C = 2 there is nothing to prove. Thus
e
by Remark (??) we may assume that F contains a primitive cube root of unity. To
simplify notations we denote the multiplication in C by (x, y) ’ xy and we put
n = NC for the norm of C. Let xi ∈ Si , i = 1, 2; we have x2 = n(xi ) = 0 by
i
Lemma (??), so that
bn (x1 x2 , x2 x1 ) = bn (x1 , x2 )2
by (??). Furthermore (x1 x2 )(x2 x1 ) = x1 (x2 x2 )x1 = 0 (by Artin™s theorem, see (??))
and
(x1 x2 )2 ’ bn (x1 x2 , 1)x1 x2 + n(x1 x2 ) · 1 = 0
implies that (x1 x2 )2 = ’bn (x1 , x2 )x1 x2 . Choosing x1 , x2 such that bn (x1 , x2 ) =
’1, we see that e1 = x1 x2 is an idempotent of C and it is easily seen that e2 =
1 ’ e1 = x2 x1 . We claim that if dim S0 = 2, then e1 = y1 y2 for any pair (y1 , y2 ) ∈
S1 — S2 such that bn (y1 , y2 ) = ’1. We have S0 = F · e1 • F · e2 if dim S0 = 2, so
that the claim will follow if we can show that bn (e1 , y1 y2 ) = 0. Let y1 = »x1 + x1
with bn (x1 , x2 ) = 0. By using (??) and the fact that n(Si ) = 0 for i = 1, 2, we
deduce
bn (e1 , y1 y2 ) = bn x1 x2 , (»x1 + x1 )y2
= n(x1 )bn (x2 , »y2 ) + bn (x1 x2 , x1 y2 )
= ’bn (x1 y2 , x1 x2 ).
However x1 x2 satis¬es
(x1 x2 )2 ’ bn (x1 x2 , 1)x1 x2 + n(x1 x2 ) = 0
hence (x1 x2 )2 = 0, since bn (x1 x2 , 1) = ’bn (x1 , x2 ) = 0. Since the algebra S0 is
´tale, we must have x1 x2 = 0 and, as claimed, bn (e1 , y1 y2 ) = 0. Similarly we have
e
e2 = y2 y1 for (y1 , y2 ) ∈ S1 — S2 such that bn (y1 , y2 ) = ’1. It follows that
e1 y1 = (1 ’ e2 )y1 = (1 ’ y2 y1 )y1 = y1 ,
y1 e1 = y1 (y1 y2 ) = 0 = e2 y1 ,
y1 e2 = y 1 , e2 y2 = y 2 = y 2 e1 and e1 y2 = 0 = y2 e2 ,
so that
S1 = { x ∈ C | e1 x = x = xe2 }
and
S2 = { x ∈ C | e2 x = x = xe1 }.
The element
e = ω 2 e1 + ωe2
is a para-unit of C• , since
e x = (ω 2 e1 + ωe2 )(ω 2i x) = ’(ωe1 + ω 2 e2 )ω 2i x = ’x
for x ∈ Si and since
e (ωe1 ’ ω 2 e2 ) = (ω 2 e1 + ωe2 )(’ω 2 e1 + ω 2 e2 ) = (’ωe1 + ω 2 e2 )
for ωe1 ’ ω 2 e2 ∈ e⊥ ‚ S0 . The claim then follows from Lemma (??).
468 VIII. COMPOSITION AND TRIALITY


34.C. Cubic separable alternative algebras. Following Faulkner [?] we
now give another approach to symmetric composition algebras over ¬elds of char-
acteristic not 3. We ¬rst recall some useful identities holding in cubic alternative
algebras. Let A be a ¬nite dimensional unital separable alternative F -algebra of
degree 3 and let
PA,a (X) = X 3 ’ TA (a)X 2 + SA (a)X ’ NA (a)1
be its generic minimal polynomial. The trace TA is linear, the form SA is quadratic
and the norm NA is cubic. As was observed in the introduction to this chapter we
have
NA (X ’ a · 1) = PA,a (X), NA (xy) = NA (x)NA (y), and TA (xy) = TA (yx).
Let
bSA (x, y) = SA (x + y) ’ SA (x) ’ SA (y),
x# = x2 ’ TA (x)x + SA (x) · 1
and
x — y = (x + y)# ’ x# ’ y # .
Note that
NA (x) = xx# = x# x.
Observe that the #-operation and the —-product are de¬ned for any cubic
algebra. They will be systematically used in Chapter IX for cubic Jordan algebras.
(34.14) Lemma. (1) NA (1) = 1, SA (1) = TA (1) = 3, 1# = 1, 1 — 1 = 2,
(2) (xy)# = y # x# ,
(3) SA (x) = TA (x# ), SA (x# ) = TA (x)NA (x), NA (x# ) = NA (x)2 ,
(4) bSA (x, y) = TA (x — y).
(5) NA (x + »y) = »3 NA (y) + »2 TA (x · y # ) + »TA (x# · y) + NA (x) for x, y ∈ A,
and » ∈ F .
(6) The coe¬cient of ±βγ in NA (±x + βy + γz) is TA x(y — z) and TA x(y — z)
is symmetric in x, y, and z.
(7) bSA (x, 1) = 2TA (x),
(8) x — 1 = TA (x) · 1 ’ x,
(9) TA (xy) = TA (x)TA (y) ’ bSA (x, y),
(10) TA (xz)y = TA x(zy) .
Proof : We may assume that F is in¬nite and identify polynomials through their
coe¬cients. NA (1) = 1 follows from the multiplicativity of NA , so that 1# = 1
and 1 — 1 = 2. Putting a = 1 in NA (X ’ a · 1) = PA,a (X) gives PA,1 (X) =
(X ’ 1)3 NA (1) = (X ’ 1)3 , hence SA (1) = TA (1) = 3.
By density it su¬ces to prove (??) for x, y such that NA (x) = 0 = NA (y).
Then (xy)# = (xy)’1 NA (xy) = y ’1 NA (y)x’1 NA (x) = y # x# .
Again by density, it su¬ces to prove (??) for x such that NA (x) = 0. We then
have NA (x ’ ») = NA (1 ’ »x’1 )NA (x). Comparing the coe¬cients of » gives (??),
and (??) follows by linearizing (??).
(??) follows by computing NA (x + »y) = NA (xy ’1 + »)NA (y).
The ¬rst claim of (??) follows by computing the coe¬cient of ±βγ in NA ±x +
(βy + γz) (and using (??)). The last claim of (??) then is clear by symmetry.
§34. SYMMETRIC COMPOSITIONS 469


(??) follows from (??), since
bSA (x, 1) = TA (x — 1) = TA x(1 — 1) = 2TA (x)
and (??) implies (??).
For (??) we have
bSA (x, y) = TA (x — y)1 = TA (x — 1)y
= TA TA (x) · 1 ’ x y = TA (x)TA (y) ’ TA (xy).
Finally, by linearizing
TA x(yx) = TA (xy)x = TA (yx2 ) = TA y x# + xTA (x) ’ SA (x)1 ,
we obtain
TA x(yz) + TA z(yx) = TA (xy)z + TA (zy)x
= TA y(x — z) + TA (yx)TA (z) + TA (yz)TA (x)
’ TA (y)bSA (x, z)
= TA y(x — z) + TA (yx)TA (z) + TA (yz)TA (x)
+ TA (xz)TA (y) ’ TA (x)TA (y)TA (z),
so that by (??) TA x(yz) + TA z(yx) = TA (xy)z + TA (zy)x is symmetric in
x, y and z. It follows that
TA x(yz) + TA z(yx) = TA y(xz) + TA z(xy)
and TA (xz)y = TA x(zy) , as claimed.
(34.15) Proposition. A cubic alternative algebra is separable if and only if the
bilinear trace form T (x, y) = TA (xy) is nonsingular.
Proof : By (??) T is associative, hence the claim follows from Dieudonn´™s Theo-
e
rem (??).
We recall:

(34.16) Proposition. For any isomorphism ± : A ’ A of cubic unital alterna-

tive algebras we have
TA ±(x) = TA (x), SA ±(x) = SA (x), NA ±(x) = NA (x).
Proof : The polynomial pA ,±(a) (X) is a minimal generic polynomial for A, hence
the claim by uniqueness.
(34.17) Theorem. Let A be a cubic separable unital alternative algebra over F of
dimension > 1. Then either :
(1) A L, for some unique (up to isomorphism) cubic ´tale algebra L over F ,
e
(2) A F —Q where Q is a unique (up to isomorphism) quaternion algebra over F ,
(3) A F — C where C is a unique (up to isomorphism) Cayley algebra over F ,
(4) A is isomorphic to a unique (up to isomorphism) central simple associative
algebra of degree 3.
In particular such an algebra has dimension 3, 5 or 9. In case (??) the generic
minimal polynomial is the characteristic polynomial, in case (??) and (??) the
product of the generic minimal polynomial pF,a (X) = X ’ a of F with the generic
minimal polynomial pC,c (X) = X 2 ’ TC (c)X + NC (c) · 1 of C = Q or C = C and
in case (??) the reduced characteristic polynomial.
470 VIII. COMPOSITION AND TRIALITY


Proof : The claim is a special case of Theorem (??).
Let 1An denote the category of central simple algebras of degree n + 1 over
F . Let I : Sepalt m’1 (2) ’ Sepalt m (3) be the functor C ’ F — C. Theorem
(??) gives equivalences of groupoids Sepalt 3 (3) ≡ S3 , Sepalt 5 (3) ≡ I(1A1 ), and
Sepalt 9 (3) I(G2 ) 1A2 .
We assume from now on (and till the end of the section) that F is a ¬eld of
characteristic di¬erent from 3. Let A be cubic alternative separable over F and let
A0 = { x ∈ A | TA (x) = 0 }.
1 1 1
Since x = 3 TA (x) · 1 + x ’ 3 TA (x) · 1 and TA x ’ 3 TA (x) = 0 we have A =
F · 1 • A0 and the bilinear trace form T : (x, y) ’ TA (xy) is nonsingular on A0 . By
Lemma (??) the polar of the quadratic form SA on A0 is ’T . Thus the restriction
of SA to A0 is a nonsingular quadratic form.
We further assume that F contains a primitive cube root of unity ω and set
µ = 1’ω . We de¬ne a multiplication on A0 by
3
1
(34.18) x y = µxy + (1 ’ µ)yx ’ 3 TA (yx)1.
This type of multiplication was ¬rst considered by Okubo [?] for matrix algebras
and by Faulkner [?] for cubic alternative algebras.
(34.19) Proposition. The algebra (A0 , ) is a symmetric composition algebra with
1
norm n(x) = ’ 3 SA (x).
Proof : The form n is nonsingular, since SA is nonsingular. We check that
1
(x y) x = x (y x) = ’ 3 SA (x)y = n(x)y.
Lemma (??) will then imply that (A0 , ) is a symmetric composition algebra. We
have 3µ(1 ’ µ) = 1. It follows that
(34.20)
(x y) x = x (y x) = µ2 xyx + (1 ’ µ)2 xyx + µ(1 ’ µ)(yx2 + x2 y)
’ 3 TA (xy)x ’ 3 µTA (xyx)1 ’ 1 (1 ’ µ)TA (xyx)1
1 1
3
= [1 ’ 2µ(1 ’ µ)]xyx + µ(1 ’ µ)(yx2 + x2 y)
’ 1 TA (xy)x ’ 3 TA (xyx)1
1
3
= 1 (xyx + yx2 + x2 y) ’ 3 TA (xy)x ’ 3 TA (xyx)1.
1 1
3

By evaluating TA on the generic polynomial, we obtain 3NA (x) = TA (x3 ) for ele-
ments in A0 . Thus
x3 + SA (x)x ’ 1 TA (x3 )1 = 0
(34.21) 3

holds for all x ∈ A0 . Since it su¬ces to prove (??) over a ¬eld extension, we may
assume that F is in¬nite. Replacing x by x + »y in (??), the coe¬cient of » must
then be zero. Hence we are lead to the identity
xyx + yx2 + x2 y ’ TA (xy)x + SA (x)y ’ TA (xyx)1 = 0
for all x, y ∈ A0 , taking into account that bSA (x, y) = ’TA (xy) on A0 . Combining
this with equation (??) shows that
(x y) x = x (y x) = ’ 1 SA (x)y = n(x)y
3
as claimed.
§34. SYMMETRIC COMPOSITIONS 471


Hence we have a functor 1 C : Sepalt m+1 (3) ’ Scomp m for m = 2, 4 and 8
given by A ’ (A0 , n). We now construct a functor 1A in opposite direction; a
straightforward computation shows that (??) is equivalent to
(34.22) xy = (1 + ω)x y ’ ωy x + bn (x, y) · 1

for the multiplication in A0 ‚ A. Thus, given a symmetric composition (S, ), it is
natural to de¬ne a multiplication (x, y) ’ x · y = xy on A = F • S by (??) for x,
y ∈ S, and by 1 · x = x = x · 1. Let 1A be the functor (S, ) ’ (F • S, ·).
(34.23) Theorem (Elduque-Myung). The functors 1 C and 1A de¬ne an equiva-
lence of groupoids
Sepalt m+1 (3) ≡ Scomp m
for m = 2, 4 and 8.
Proof : We ¬rst show that A = 1A(S) = F • S is a separable alternative algebra of
degree 3: Let x = ±1 + a, ± ∈ F and a ∈ S. We have
x2 = ±2 + bn (a, a) 1 + 2±a + a a
and
xx2 = x2 x = [±3 + 3±bn (a, a) + bn (a a, a)]1 + [3±2 + 3n(a)]a + 3±(a a).
It follows that
x3 ’ 3±x2 + 3±2 ’ 3n(a) x = [±3 ’ 3n(a)± + bn (a a, a)]1
so that elements of A satisfy a polynomial condition of degree 3
pA,x (X) = X 3 ’ TA (x)X 2 + SA (x)X ’ NA (x)1 = 0
with
SA (x) = 3±2 ’ 3n(a)
TA (x) = 3±,

and

NA (x) = ±3 ’ 3±n(a) + bn (a a, a)
for x = ±1 + a. To show that A is of degree 3 we may assume that the ground
¬eld F is in¬nite and we need an element x ∈ A such that the set {1, x, x2 } is
linearly independent. Because x2 = x x + (x, x)1 for x ∈ S, it su¬ces to have
x ∈ S such that {1, x, x x} is linearly independent. Since TA (1) = 3, while
TA (x) = 0 = TA (x x), the only possible linear dependence is between x and x x.
If {x, x x} is linearly dependent for all x ∈ S, there is a map f : S ’ F such
that x x = f (x)x for x ∈ S. By the following Lemma (??) f is linear. Since
x x x = n(x)x we get n(x) = f (x)2 . This is only possible if dimF S = 1. We
next check that A is alternative. It su¬ces to verify that
a2 b = a(ab) and ba2 = (ba)a for a, b ∈ S.
We have
a2 b = [a a + (a, a)]b = (1 + ω)(a a) b ’ ωb (a a) + bn (a a, b)
472 VIII. COMPOSITION AND TRIALITY


and
a(ab) = a[(1 + ω)a b ’ ωb a + bn (a, b)]
= (1 + ω)[a (a b) ’ ω(a b) a + bn (a, a b)]
’ ω[(1 + ω)a (b a) ’ ω(b a) a + bn (a, b a)] + bn (a, b)a.
By (??) we have
(a a) b + (b a) a = bn (a, b)a = b (a a) + a (a b).
This, together with the identities
bn (a, a b) = bn (a a, b) = bn (b, a a) = bn (b a, a) = bn (a, b a)
which follow from the associativity of n, implies that a2 b = a(ab). The proof of
ba2 = (ba)a is similar. Thus A is alternative of degree 3. We next check that A is
separable. We have for x = ± + a, y = β + b,
xy = ±β + βa + ±b + (1 + ω)a b ’ ωb a + bn (a, b)
so that
T (x, y) = TA (xy) = 3±β + 3bn (a, b)
is a nonsingular bilinear form. Since the trace form of a cubic alternative algebra is
associative (Lemma (??)), A is separable by Dieudonn´™s Theorem (??). We ¬nally
e
have an equivalence of groupoids since
(F • S)0 = S and F • A0 = A
and since formulas (??) and (??) are equivalent.

(34.24) Lemma. Let F be an in¬nite ¬eld and let (S, ) be an F -algebra. If there
exists a map f : S ’ F such that x x = f (x)x for all x ∈ S, then f is linear.
Proof : (Elduque) If S is 1-dimensional, the claim is clear. So let (e1 , . . . , en ) be a
xi ei and ei ej = k ak ek , we have i,j ak xi xj = f (x)xk
basis of S. For x = ij ij
’1
for any k. Thus f (x) = gk (x)xk for some quadratic homogeneous polynomial
gk (x) and k = 1, . . . , n in the Zariski open set
D(xk ) = { x ∈ S | xk = 0 }.
For any pair i, j we have gi (x)xj = gj (x)xi in D(xi ) © D(xj ), so by density
gi (x)xj = gj (x)xi holds for any x ∈ S. Unique factorization over the polyno-
mial ring F [x1 , . . . xn ] shows that there exists a linear map φ : S ’ F such that
gi (x) = xi φ(x). It is clear that f = φ.

(34.25) Remark. Let A be central simple of degree 3 over F and assume that F
has characteristic di¬erent from 3 and that F contains a primitive cube root of
unity. The form n from (??) is then hyperbolic on A0 : by Springer™s Theorem
(see [?, p. 119]) we may assume that A is split, and in that case the claim is easy
to check directly. Hence, if A and A are of degree 3 and are not isomorphic, the
0
compositions (A0 , ) and (A , ) are nonisomorphic (by (??)) but have isometric
norms. This is in contrast with Cayley (or para-Cayley) composition algebras.
§34. SYMMETRIC COMPOSITIONS 473


(34.26) Remark. The polar of a cubic form N is
N (x, y, z) = N (x + y + z) ’ N (x + y) ’ N (x + z) ’ N (y + z)
+ N (x) + N (y) + N (z)
and N is nonsingular if its polar is nonsingular, i.e., if N (x, y, z) = 0 for all x, y im-
plies that z = 0. Let A be an F -algebra. If char F = 2, 3 a necessary and su¬cient
condition for A to admit a nonsingular cubic form N which admits composition (i.e.,
such that N (xy) = N (x)N (y)) is that A is cubic separable alternative and N NA
(see Schafer [?, Theorem 3]). Thus, putting x = ± · 1 + a ∈ F · 1 • A0 , the multi-
plicativity of NA (x) = ±3 ’ 3±n(a) + bn (a a, a) for the multiplication (x, y) ’ xy
of A is equivalent by Proposition (??) to the multiplicativity of n = ’ 1 SA for the
3
multiplication (a, b) ’ a b of A0 . It would be nice to have a direct proof!
A symmetric composition algebra isomorphic to a composition (A0 , ) for A
central simple of degree 3 is called an Okubo composition algebra or a composition
algebra of type 1A2 since its automorphism group is a simple adjoint algebraic group
of type 1A2 . Twisted forms of Okubo algebras are again Okubo algebras. The
groupoid of Okubo composition algebras over a ¬eld F containing a primitive cube
root of unity is denoted 1Oku. We have an equivalence of groupoids 1Oku ≡ 1A2 (if
F contains a primitive cube root of unity).
For para-Hurwitz compositions of dimension 4 or 8 we have the following situ-
ation:
(34.27) Proposition. Let I : Hurw m ’ Sepalt m+1 (3) be the functor C ’ F — C,
P : Hurw m ’ Hurw m the para-Hurwitz functor and J : Hurw m ’ Scomp m the
inclusion. Then the map
·C : C ’ (F — C)0 z ’ TC (z), ωz + ω 2 z
given by
is a natural transformation between the functors 1 C —¦ I and J —¦ P, i.e., the diagram
1
C
Sepalt m+1 (3) ’ ’ ’ Scomp m
’’
¦ ¦
I¦ ¦J
P
’ ’ ’ Hurw m
’’
Hurw m
commutes up to ·C .
Proof : It su¬ces to check that ·C is an isomorphism of the para-Hurwitz algebra
(C, ) with the symmetric composition algebra (F — C)0 , . We shall use that
TA (x) = ξ + TC (c), SA (x) = NC (c) + ξTC (c) and NA (x) = ξNC (c) for A = F — C,
ξ ∈ F , c ∈ C and TC the trace and NC the norm of C. If char F = 2, we decompose
C = F · 1 • C 0 and set u = (2, ’1) ∈ A0 . We then have
·C (βe + x) = βu + (1 + 2ω)x.
The element u satis¬es u u = u and (0, x) u = u (0, x) = (0, ’x) for x ∈ C 0 .
Thus it su¬ces to check the multiplicativity of ·C on products of elements in C 0 ,
in which case the claim follows by a tedious but straightforward computation. If
char F = 2, we choose v ∈ C with TC (v) = 1, to have C = F · v • C 0 . We then
have
·C (βv + x) = (β, βv + x + ω 2 β)
474 VIII. COMPOSITION AND TRIALITY

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