if L is a ¬eld we may replace L by L — L. Then (V, Q) is similar to a Cayley

composition (V0 , Q0 ) with

Q0 = 1 ⊥ δ — (C, n)0

(see Theorem (??) and Lemma (??)). Since f3 (J) is the cohomological invariant of

the norm of C, we get (??).

(??) ’ (??) We may assume that J is a division algebra. (??) also implies

(??) and a reduced algebra with f3 = 0 is split. Let x ∈ V with Q(x) = 0. We

have Q β(x) = 0 and Q x, β(x) = 0 (by Proposition (??) and Theorem (??),

since J is a division algebra). Thus U = Lx • Lβ(x) is a 2-dimensional twisted

composition. By Proposition (??), Q|U is the trace of a hermitian 1-form over L—K

for some quadratic ´tale F -algebra K. Furthermore K is split if Q|U is isotropic.

e

Now the Springer construction J1 = L • U is a 9-dimensional Freudenthal algebra

and the Witt index of the bilinear trace form of J1 is at least 2. As shown in

Example (??), (??), J1 is a second Tits construction J1 = J(L — K, 1 — ιK , u1 , ν1 )

and by Example (??), (??), J1 H(B1 , „1 ) where B1 is central simple of degree 3

with a unitary involution „1 . Moreover the center K1 of B1 is the discriminant

algebra ∆(L) (since K as above is split). Since the trace on H(B1 , „1 ) is of Witt

index ≥ 2, „1 is distinguished (Proposition (??)). Furthermore, by Theorem (??),

J is a second Tits construction J = J(B1 , „1 , u, ν) for the given (B1 , „1 ). We

have f3 Int(u) —¦ „1 = f3 (J) and since (??) implies (??), „ = Int(u) —¦ „1 is also

„ and there exist » ∈ F — and w ∈ B — such

distinguished. By Theorem (??) „

that u = »w„ (w). By Lemma (??) we may assume that u = » ∈ F — . Then the

Tits construction J2 = J(L—K1 , 1—ιK1 , », ν) is a subalgebra of J = J(B1 , „1 , u, ν).

By Example (??) of (??), J2 H(B2 , „2 ) and the center of B2 is K1 — ∆(L). Since

K1 ∆(L), J2 (A — Aop , exchange) and we conclude using Theorem (??).

(40.6) Remark. The equivalence of (??) and (??) in (??) is due to Petersson-

Racine [?, Theorem 4.7] if F contains a primitive cube root of unity. The trace

form then has maximal Witt index.

(40.7) Proposition. Let J be an Albert algebra. The following conditions are

equivalent:

(1) J = J(B, „, u, ν) is a second Tits construction with „ a distinguished unitary

involution of B.

(2) The Witt index w(T ) of the bilinear trace form T of J is at least 8.

(3) f5 (J) = 0.

Proof : We use the notations of the proof of Lemma (??).

(??) ’ (??) The bilinear form bK/F — b, c is hyperbolic if „ is distinguished.

Thus T„,„ has Witt index at least 6. By Proposition (??), T„ has Witt index at

least 2, hence the claim.

(??) ’ (??) If w(T ) ≥ 8, a, e, f is isotropic, hence f5 (J) = 0.

The proof of (??) ’ (??) goes along the same lines.

(??) ’ (??) We assume that J is a division algebra. Let J = L • V be a

Springer decomposition of J; since (??) ’ (??) holds, we get that T |V is isotropic.

We may choose x ∈ V such that T (x, x) = 0 and such that U = Lx • β(x) is a 2-

dimensional twisted composition. Then J1 = L•U is a Freudenthal subalgebra of J

§40. COHOMOLOGICAL INVARIANTS 535

of dimension 9, hence of the form H(B, „ ). Since w(T |J1 ) ≥ 2, „ is distinguished.

The claim then follows from Theorem (??).

We now indicate how one can associate a third cohomological invariant g 3 (J)

to an Albert algebra J. We refer to Rost [?], for more information (see also the Reference

paper [?] of Petersson and Racine for an elementary approach). By Theorem (??), missing: connect

we may assume that J = J(B, „, u, ν) is a second Tits process and by Lemma (??) to H3

√

that NrdB (u) = νι(ν) = 1. Let Lν be the descent of K( 3 ν) under the action given

√

by ιK on K and ξ ’ ξ ’1 for ξ = 3 ν. Then Lν de¬nes a class in H 1 (F, µ3[K] )

by Proposition (??). On the other hand, the algebra with involution (B, „ ) de-

termines a class g2 (B, „ ) ∈ H 2 (F, µ3[K] ) by Proposition(??). Since there exists a

canonical isomorphism of Galois modules µ3[K] — µ3[K] = Z/3Z (with trivial Galois

action on Z/3Z), the cup product g2 (B, „ ) ∪ [ν] de¬nes a cohomology class g3 in

H 3 (F, Z/3Z). If K = F — F , B = A — Aop and ν = (», »)’1 , then [A] ∈ H 2 (F, µ3 ),

[»] ∈ H 1 (F, µ3 ) and we have g3 = [A] ∪ [»] ∈ H 3 (F, Z/3Z). The following result is

due to Rost [?]:

(40.8) Theorem. (1) The cohomology class g3 ∈ H 3 (F, Z/3Z) is an invariant of

the Albert algebra J = J(B, „, u, ν), denoted g3 (J).

(2) We have g3 (J) = 0 if and only if J has zero divisors.

(40.9) Remark. By de¬nition we have g3 = g2 (B, „ )∪[ν] and by Proposition (??)

we know that g2 (B, „ ) = ± ∪ β with ± ∈ H 1 (F, Z/3Z[K] ) and β ∈ H 1 (F, µ3 ); thus

g3 ∈ H 1 (F, Z/3Z[K] ) ∪ H 1 (F, µ3 ) ∪ H 1 (F, µ3[K] )

is a decomposable class.

It is conjectured that the three invariants f3 (J), f5 (J) and g3 (J) classify

Freudenthal algebras of dimension 27. This is the case if g3 = 0; then J is re-

duced, J H3 (C, ±), f3 (J) = f3 (C) determines C, f3 (J), f5 (J) determine the trace

and the claim follows from Theorem (??).

Theorem (??) allows to prove another part of the converse to Lemma (??)

which is due to Petersson-Racine [?, p. 204]:

(40.10) Proposition. If J(B, „, u, ν) J(B, „, u , ν ) then ν = ν Nrd(w) for

some w ∈ B — .

Proof : The claim is clear if B is not a division algebra, since then the reduced

norm map is surjective. Assume now that J = J(B, „, u, ν) J = J(B, „, u , ν ).

By (??), (??), we may assume that NK/F (ν) = 1 = NK/F (ν ). Let L, resp. L , be

the cubic extensions of F determined by ν, resp. ν , as in Proposition (??). We have

[B] ∪ [L] = g3 (J) = g3 (J ) = [B] ∪ [L ], hence [B] ∪ ([L ][L]’1 ) = 0 in H 3 (F, Z/3Z).

The class [L ][L]’1 comes from ν ν ’1 . Since (u, ν ν ’1 ) is obviously admissible we

have a Tits construction J = J(B, „, u, ν ν ’1 ) whose invariant g3 (J ) is zero. By

Theorem (??), (??), J has zero divisors and by Theorem (??) ν ν ’1 is a norm

of B.

Let J(A, ») be a ¬rst Tits construction. Since an admissible pair for this

construction can be assumed to be of the form 1, (», »’1 ) we get

(40.11) Corollary. Let A be a central division algebra of degree 3 and let », » ∈

F — . The Albert algebras J(A, ») and J(A, » ) are isomorphic if and only if » »’1 ∈

NrdA (A— ).

536 IX. CUBIC JORDAN ALGEBRAS

We now prove the result of Parimala, Sridharan and Thakur [?] quoted in

Remark (??).

(40.12) Theorem. Let (B, „ ) be a degree 3 central simple K-algebra with a unitary

involution. Then J(B, „, u1 , ν1 ) J(B, „, u2 , ν2 ) if and only if there exists some

—

w ∈ B such that u2 = wu1 „ (w) and ν2 = ν1 NrdB (w).

Proof : Let (u1 , ν1 ), (u2 , ν2 ) be admissible pairs. Recall from (??) the equiva-

lence relation ≡ on admissible pairs. Assume that J(B, „, u1 , ν1 ) J(B, „, u2 , ν2 ).

By (??), we have some u3 such that (u1 , ν1 ) ≡ (u3 , ν2 ) and by (??) (u3 , ν2 ) ≡

u2 , NrdB (w)’1 ν2 for some w ∈ B — such that u2 = wu3 „ (w). One has NrdB (u3 ) =

ν2 ν 2 = NrdB (u3 ) since (u3 , ν2 ) and (u2 , ν2 ) are admissible pairs, thus »» = 1 for

» = NrdB (w). Let „2 = Int(u2 ) —¦ „ . By the next lemma applied to „2 , there exists

w ∈ B — such that w „2 (w ) = 1 and » = NrdB (w ). It follows from w „2 (w ) = 1

that w u2 „ (w ) = u2 , hence

u2 , NrdB (w)’1 ν2 ≡ u2 , NrdB (w ) NrdB (w)’1 ν2 = (u2 , ν2 )

and (u1 , ν1 ) ≡ (u2 , ν2 ) as claimed. The converse is (??), (??).

(40.13) Lemma (Rost). Let (B, „ ) be a degree 3 central simple K-algebra with a

unitary involution. Let w ∈ B — be such that » = NrdB (w) ∈ K satis¬es »» = 1.

Then there exists w ∈ B — such that w „ (w ) = 1 and » = NrdB (w ).

Proof : Assume that an element w as desired exists and assume that M = K[w ] ‚

B is a ¬eld. We have „ (M ) = M , so let H = M „ be the sub¬eld of ¬xed elements

under „ . The extension M/H is of degree 2 and by Hilbert™s Theorem 90 (??) we

√

may write w = u„ (u)’1 . Since M √ H — K and K = F ( a) for some a ∈ F — , we

=

may choose u of the form u = h + a with h ∈ H. Then

√ √

» = Nrd(w ) = NrdB (h + a) NrdB (h ’ a)’1 .

On the other hand » = y„ (y)’1 by Hilbert™s Theorem 90 (??), so that h ∈ H(B, „ )

is a zero of

√ √ √ ’1

•(h) = y NrdB (h ’ a) ’ y NrdB (h + a) a .

√ ’1

(the factor a is to get an F -valued function on H(B, „ )). Reversing the argu-

√ √

ment, if • is isotropic on H(B, „ ), then w = (h + a)(h ’ a)’1 is as desired. The

function • is polynomial of degree 3 and it is easily seen that • is isotropic over K.

It follows that • is isotropic over F (see Exercise ?? of this chapter).

(40.14) Remark. Suresh has extended Lemma (??) to algebras of odd degree

with unitary involution (see [?], see also Exercise 12, (b), in Chapter III).

Theorem (??) has a nice Skolem-Noether type application, which is also due

to Parimala, Sridharan and Thakur [?]:

(40.15) Corollary. Let (B1 , „1 ), (B2 , „2 ) be degree 3 central simple algebras over K

with unitary involutions. Suppose that H(B1 , „1 ) and H(B2 , „2 ) are subalgebras of

an Albert algebra J and that ± : H(B1 , „1 ) H(B2 , „2 ) is an isomorphism of Jordan

algebras. Then ± extends to an automorphism of J.

Proof : In view of Theorem (??), (??), we have isomorphisms

∼ ∼

ψ1 : J(B1 , „1 , u1 , ν1 ) ’ J,

’ ψ2 : J(B2 , „2 , u2 , ν2 ) ’ J.

’

§41. EXCEPTIONAL SIMPLE LIE ALGEBRAS 537

∼

By Proposition (??) ± extends to an isomorphism ± : (B1 , „1 ) ’ (B2 , „2 ), thus we

’

get an isomorphism of Jordan algebras

∼

J(±) : J(B1 , „1 , u1 , ν1 ) ’ J B2 , „2 , ±(u1 ), ν1 .

’

But J B2 , „2 , ±(u1 ), ν1 J(B2 , „2 , u2 , ν2 ), since both are isomorphic to J. By The-

—

orem (??), there exists w ∈ B2 such that u2 = w±(u1 )„2 (w) and ν2 = NrdB (w)ν1 .

Let

∼

φ : J B2 , „2 , ±(u1 ), ν1 ’ J(B2 , „2 , u2 , ν2 )

’

be given by (a, b) ’ (a, bw). Then φ restricts to the identity on H(B2 , „2 ) and

’1

ψ = ψ2 —¦ φ —¦ J(±) —¦ ψ1

is an automorphism of J extending ±.

40.A. Invariants of twisted compositions. Let F be a ¬eld of character-

istic not 2. To a twisted composition (V, L, Q, β) we may associate the following

cohomological invariants:

(a) a class f1 = [δ] ∈ H 1 (F, µ2 ), which determines the discriminant ∆ of L;

(b) a class g1 ∈ H 1 F, (Z/3Z)δ which determines L (with the ¬xed discrimi-

nant ∆ given by the cocycle δ);

(c) invariants f3 ∈ H 3 (F, µ2 ), f5 ∈ H 5 (F, µ2 ), and g3 ∈ H 3 (F, Z/3Z) which

are the cohomological invariants associated with the Freudenthal algebra

J(L, V ) (see Theorem (??)).

As for Freudenthal algebras, it is unknown if these invariants classify twisted com-

positions, however:

(40.16) Proposition. The invariant g3 of a twisted composition (V, L, Q, β) is

trivial if and only if (V, L, Q, β) is similar to a composition “(C, L) of type G2 , in

which case (V, L, Q, β) is classi¬ed up to similarity by f1 and g1 (which determine L)

and by f3 (which determines C).

Proof : By Theorem (??) J(V, L) has zero divisors if and only if (V, L, Q, β) is

similar to a composition of type G2 , hence the claim by Theorem (??).

§41. Exceptional Simple Lie Algebras

There exists a very explicit construction, due to Tits [?], of models for all

exceptional simple Lie algebras. This construction is based on alternative algebras

of degree 2 or 1 and Jordan algebras of degree 3 or 1. We sketch it and refer to

[?], to the book of Schafer [?] or to the notes of Jacobson [?] for more details. We

assume throughout that the characteristic of F is di¬erent from 2 and 3. Let A,

B be Hurwitz algebras over F and let J = H3 (B, ±) be the Freudenthal algebra

associated to B and ± = diag(±1 , ±2 , ±3 ). As usual we write A0 , resp. J 0 for the

trace zero elements in A, resp. J. We de¬ne a bilinear product — in A0 by

1

a — b = ab ’ 2 T (a, b)

where T (a, b) = TA (ab), a, b ∈ A, is the bilinear trace form of A. Let a , resp. ra ∈

EndF (A) be the left multiplication map a (x) = ax, resp. the right multiplication

map ra (x) = xa. For f , g ∈ EndF (A) we put [f, g] = f —¦ g ’ g —¦ f for the Lie

product in EndF (A). It can be checked that in any alternative algebra A

Da,b = [ a , b ] + [ a , rb ] + [ra , rb ]

538 IX. CUBIC JORDAN ALGEBRAS

is a derivation. Similarly we may de¬ne a product on J 0 :

x — y = xy ’ 1 T (x, y)

3

where T (x, y) = TJ (xy). We now de¬ne a bilinear and skew-symmetric product

[ , ] on the direct sum of F -vector spaces

L(A, J) = Der(A, A) • A0 — J 0 • Der(J, J)

as follows:

(1) [ , ] is the usual Lie product in Der(A, A) and Der(J, J) and satis¬es [D, D ] = 0

for D ∈ Der(A, A) and D ∈ Der(J, J),

(2) [a — x, D + D ] = D(a) — x + a — D (x) for a ∈ A0 , x ∈ J 0 , D ∈ Der(A, A) and

D ∈ Der(J, J),

1 1

(3) [a — x, b — y] = 12 T (x, y)Da,b + (a — b) — (x — y) + 2 T (a, b)[rx , ry ] for a, b ∈ B 0

and x, y ∈ J 0 .

With this product L(A, J) is a Lie algebra. As A and B vary over the possible

composition algebras the types of L(A, J) can be displayed in a table, whose last

four columns are known as Freudenthal™s “magic square”:

dim A F F —F —F H3 (F, ±) H3 (K, ±) H3 (Q, ±) H3 (C, ±)

1 0 0 A1 A2 C3 F4

2 0 A2 A2 • A 2 A5 E6

U

4 A1 A1 • A 1 • A 1 C3 A5 D6 E7

8 G2 D4 F4 E6 E7 E8

Here K stands for a quadratic ´tale algebra, Q for a quaternion algebra and C for a

e

Cayley algebra; U is a 2-dimensional abelian Lie algebra. The fact that D4 appears

in the last row is one more argument for considering D4 as exceptional.

Exercises

1. (Springer [?, p. 63]) A cubic form over a ¬eld is isotropic if and only if it is

isotropic over a quadratic extension.

2. For any alternative algebra A over a ¬eld of characteristic not 2, A + is a special

Jordan algebra.

3. Show that in all cases considered in §??, §??, and §?? the generic norm

NJ ( i xi ui ) is irreducible in F [x1 , . . . , xn ].

4. Let C be a Hurwitz algebra. Show that H2 (C, ±) is the Jordan algebra of a

quadratic form.

5. Show that a Jordan division algebra of degree 2 is the Jordan algebra J(V, q)

of a quadratic form (V, q) such that bq (x, x) = 1 for all x ∈ V .

6. Let J be a cubic Jordan structure. The following conditions are equivalent:

(a) J contains an idempotent (i.e an element e with e2 = e) such that SJ (e) =

1.

(b) J contains a nontrivial idempotent e.

(c) J contains nontrivial zero divisors.

(d) There is some nonzero a ∈ J such that NJ (a) = 0.

(e) There is some nonzero a ∈ J such that a# = 0.

7. (a) Let A be a cubic separable alternative algebra and let » ∈ F — . Check that

the norm NA induces a cubic structure J(A, ») on A • A • A.

EXERCISES 539

(b) Show that J(A, ») H3 (C, ±) for some ± if A = F — C, with C a Hurwitz

algebra over F .

8. (Rost) Let A be central simple of degree 3, » ∈ F — , and J = J(A, ») the

corresponding ¬rst Tits construction. Put:

V (J) = { [ξ] ∈ P26 | ξ ∈ J, ξ # = 0 }

F

PGL1 (A) = { x ∈ P(A) = P8 | NrdA (x) = 0 }

F

SL1 (A)» = { x ∈ A | NrdA (x) = » }.

Show that

(a) V (J) is the projective variety with coordinates [a, b, c] ∈ P(A • A • A) and

equations

a# = bc, b# = »’1 ca, c# = »ab

and V (J) is smooth.

(b) The open subvariety U of V (J) given by

NrdA (a) NrdA (b) NrdA (c) = 0

is parametrized by coordinates

[a, b] ∈ P(A • A)

with NrdA (a) = » NrdA (b) and NrdA (a) NrdA (b) = 0.

(c) SL1 (A)» — PGL1 (A) is an open subset of V (J).

(d) SL1 (A)—PGL1 (A) and SL3 (F )—PGL3 (F ) are birationally equivalent (and

rational). Hint: Use that J(A, 1) J M3 (F ), 1 .

Show that a special Jordan central division algebra over R is either isomor-

9.

phic to R or to to the Jordan algebra of a negative de¬nite quadratic form of

dimension ≥ 2 over R.

10. Let J be an Albert algebra over F . Show that:

(a) J is split if F is ¬nite or p-adic.

(b) J is reduced if F = R or if F is a ¬eld of algebraic numbers (Albert-

Jacobson [?]).

Let Ca be the nonsplit Cayley algebra over R. Show that the Albert algebras

11.

H3 (Ca , 1), H3 Ca , diag(1, ’1, 1) , and H3 Cs , diag(1, ’1, 1) are up to isomor-

phism all Albert algebras over R.

12. Let F be a ¬eld of characteristic not 2 and J = H3 (C, 1), J1 = H3 (Q, 1),

J2 = H3 (K, 1) and J3 = H3 (F, 1) for C a Cayley algebra, Q a quaternion

algebra, and K = F (i), i2 = a, a quadratic ´tale algebra. Show that

e

(a) AutF (J/J1 ) SL1 (Q).

(b) AutF (J/J2 ) SU(M, h) where M = K ⊥ ‚ C (with respect to the norm)

and

h(x, y) = NC (x, y) + a’1 iNC (ix, y).

In particular AutF (J/J2 ) SL3 (F ) if K = F — F .

(c) AutF (J/J3 ) AutF (C).

(d) AutF (J1 ) — SL1 (Q) ’ AutF (J) (“C3 — A1 ‚ F4 ”).

(e) AutF (J2 ) — SU(M, h) ’ AutF (J) (“A2 — A2 ‚ F4 ”).

(f) AutF (J3 ) — SL2 (F ) ’ AutF (J) (“G2 — A1 ‚ F4 ”).

540 IX. CUBIC JORDAN ALGEBRAS

(g) Let

±«

00 0

V = 0 x c x ∈ F, c ∈ C ‚ J.

0 c ’x

Show that AutF (J/F · E11 ) Spin9 (V, T |V ). (“B4 ‚ F4 ”).

(h) AutF (J/F · E11 • F · E22 • F · E33 ) Spin(C, n).

Observe that (??), (??), and (??) give the possible types of maximal subgroups

of maximal rank for F4 .

13. (Parimala, Sridharan, Thakur) Let J = J(B, „, uν) be a second Tits construc-

tion with B = M3 (K) and u ∈ B such that NrdB (u) = 1. Let u K be the

hermitian form of rank 3 over K determined by u and let C = C( u K , K) be

the corresponding Cayley algebra, as given in Exercise ?? of Chapter ??. Show

that the class of C is the f3 -invariant of J.

Notes

§??. The article of Paige [?] provides a nice introduction to the theory of

Jordan algebras. Jacobson™s treatise [?] gives a systematic presentation of the

theory over ¬elds of characteristic not 2. Another important source is the book

of Braun-Koecher [?] and a forthcoming source is a book by McCrimmon [?]. If

2 is not invertible the Jordan identity (??) is unsuitable and a completely new

characteristic-free approach was initiated by McCrimmon [?]. The idea is to ax-

1

iomatize the quadratic product aba instead of the Jordan product a q b = 2 (ab + ba).

McCrimmon™s theory is described for example in Jacobson™s lecture notes [?] and

[?]. Another approach to Jordan algebras based on an axiomatization of the notion

of inverse is provided in the book of Springer [?]. The treatment in degree 3 is

similar to that given by McCrimmon for exceptional Jordan algebras (see [?, §5]).

A short history of Jordan algebras can be found in Jacobson™s obituary of Albert

[?], and a survey for non-experts is given in the paper by McCrimmon [?]. For

more recent developments by the Russian School, especially on in¬nite dimensional

Jordan algebras, see McCrimmon [?].

A complete classi¬cation of ¬nite dimensional simple formally real Jordan al-

gebras over R appears already in Jordan, von Neumann and Wigner [?]35 . They

conjectured that H3 (C, 1) is exceptional and proposed it as a problem to Albert.

Albert™s solution appeared as a sequel [?] to their paper. Much later, Albert again

took up the theory of Jordan algebras; in [?] he described the structure of simple

Jordan algebras over algebraically closed ¬elds of characteristic not 2, assuming

that the algebras admit an idempotent di¬erent from the identity. (The existence

of an identity in a simple ¬nite dimensional Jordan algebra was showed by Albert

in [?].) The gap was ¬lled by Jacobson in [?]. In [?] Schafer showed that reduced

exceptional Jordan algebras of dimension 27 are all of the form H3 (C, ±). A system-

atic study of algebras H3 (C, ±) is given in Freudenthal™s long paper [?], for example

the fact that they are of degree 3. In the same paper Freudenthal showed that

the automorphism group of a reduced simple exceptional Jordan algebra over R is

of type F4 by computing the root system explicitly. In a di¬erent way, Springer

35 A a2 = 0 implies every ai = 0.

Jordan algebra is said to be formally real if i

NOTES 541

[?, Theorem 3] or [?], showed that the automorphism group is simply connected

of dimension 52, assuming that F is a ¬eld of characteristic di¬erent from 2 and

3, and deduced its type using the classi¬cation of simple algebraic groups. The

fact that the derivation algebra of an exceptional Jordan algebra is a Lie algebra

of type F4 can already be found in Chevalley-Schafer [?]. Here also it was assumed

that the base ¬eld has characteristic di¬erent from 2 and 3. Observe that this Lie

algebra is not simple in characteristic 3. Split simple groups of type E6 also occur in

connection with simple split exceptional Jordan algebras, namely as automorphism

groups of the cubic form N , see for example Chevalley-Schafer [?], Freudenthal [?]

and Jacobson [?].

The structure of algebras H3 (C, ±) over ¬elds of characteristic not 2, 3 was sys-

tematically studied by Springer in a series of papers ([?], [?], [?], and [?]). Some of

the main results are the fact that the bilinear trace form and C determine H3 (C, ±)

(Theorem (??), see [?, Theorem 1, p. 421]) and the fact that the cubic norm deter-

mines C (see [?, Theorem 1]). Thus the cubic norm and the trace form determine

the algebra. The fact that the trace form alone determines the algebra was only

recently noticed by Serre and Rost (see [?, § 9.2]). The fact that the isomorphism

class of C is determined by the isomorphism class of H3 (C, ±) is a result due to

Albert-Jacobson [?]. For this reason C is usually called the coordinate algebra of

H3 (C, ±). A recent survey of the theory of Albert algebras has been given by

Petersson and Racine [?].

It is unknown if a division Albert algebra J always contains a cyclic cubic

¬eld extension (as does an associative central simple algebra of degree 3). However

this is true (Petersson-Racine [?, Theorem 4]) if char F = 3 and F contains a

primitive cube root of unity: it su¬ces to show that J contains a Kummer extension

F [X]/(X 3 ’ »), hence that SJ restricted to J 0 = { x ∈ J | TJ (x) = 0 } is isotropic.

In view of Springer™s theorem, we may replace J by J — L where L is a cubic ´tale e

0

subalgebra of J. But then J is reduced and then SJ |J is isotropic.

§??. There are a number of characterizations of cubic Jordan algebras. One

is due to Springer [?], assuming that char F = 2, 3: Let J be a ¬nite dimensional

F -algebra with 1, equipped with a quadratic form Q such that

(a) Q(x)2 = Q(x2 ) if bQ (x, 1) = 0,

(b) bQ (xy, z) = bQ (x, yz),

3

(c) Q(1) = 2 .

1

Then J is a cubic Jordan algebra and Q(x) = 2 TJ (x2 ).

The characterization we use in §?? was ¬rst suggested by Freudenthal [?] and

was established by Springer [?] for ¬elds of characteristic not 2 and 3. We follow the

description of McCrimmon [?], which is systematically used by Petersson-Racine

in their study of cubic Jordan algebras (see for example [?] and [?]). The Springer