x ∈ U.

(3) The identities

(a) SJ (1) = TJ (1) = 3, NJ (1) = 1, 1# = 1,

(b) SJ (x) = TJ (x# ), bSJ (x, y) = TJ (x — y),

(c) bSJ (x, 1) = 2TJ (x),

(d) 2SJ (x) = TJ (x)2 ’ TJ (x2 ),

(e) TJ (x — y) = TJ (x)TJ (y) ’ T (x, y),

(f) x## = NJ (x)x,

(g) T (x — y, z) = T (x, y — z)

hold in J.

§38. CUBIC JORDAN ALGEBRAS 519

Proof : (??) can be directly checked. For (??) we observe that F [x] is ´tale if and

e

only if the generic minimal polynomial PJ,x of x has pairwise distinct roots (in an

algebraic closure) i.e., the discriminant of PJ,x (as a function of x) is not zero. This

de¬nes the open set U . It can be explicitely shown that the set U is non-empty

if J is reduced, i.e., is not a division algebra. Thus we may assume that J is a

division algebra. Then, by the following lemma (??), F is in¬nite. Again by (??)

J is reduced over an algebraic closure Falg of F . The set U being non-empty over

Falg and F being in¬nite, it follows that U is non-empty. We refer to [?] or [?]

for (??). A proof for cubic alternative algebras is in (??).

An element x ∈ J is invertible if it is invertible in the algebra F [x] ‚ J.

We say that a cubic norm structure is a division cubic norm structure if every

nonzero element has an inverse. Such structures are (non-reduced) Freudenthal

algebras and can only exist in dimensions 1, 3, 9, and 27. In dimension 3 we get

separable ¬eld extensions and in dimension 9 central associative division algebras of

degree 3 or symmetric elements in central associative division algebras of degree 3

over quadratic separable ¬eld extensions, with unitary involutions. Corresponding

examples in dimension 27 will be given later using Tits constructions.

(38.3) Lemma. An element x ∈ J is invertible if and only if NJ (x) = 0 in F . In

that case we have x’1 = NJ (x)’1 x# . Thus a cubic norm structure J is a division

cubic norm structure if and only if NJ (x) = 0 for x = 0 in J, i.e., NJ is anisotropic.

In particular a cubic norm structure J of dimension > 3 is reduced (i.e., is not a

division algebra) if F is ¬nite or algebraically closed.

Proof : If NJ (x) = 0 for x = 0, we have by Lemma (??) x## = NJ (x)x = 0 hence

either u = x# or u = x satis¬es u# = 0 and u = 0. We then have SJ (u) =

TJ (u# ) = 0 so that u satis¬es

0 = u# = u2 ’ TJ (u)u = 0.

If TJ (u) = 0 we have u2 = 0; if TJ (u) = 0 we may assume that TJ (u) = 1 and

u2 = u, however u = 1. Thus in both cases u is not invertible (see also Exercise ??

of this chapter). The claim for F ¬nite or algebraically closed follows from the fact

that such a ¬eld is Ci , i ¤ 1 (see for example the book of Greenberg [?, Chap. 2]

or Scharlau [?, § 2.15]). Thus NJ , which is a form of degree 3 in 9 or 27 variables

cannot be anisotropic over a ¬nite ¬eld or an algebraically closed ¬eld.

(38.4) Proposition. If char F = 2, the categories Cubjord and Sepjord(3) are

isomorphic.

Proof : Any separable cubic Jordan algebra determines a cubic norm structure and

an isomorphism of separable cubic Jordan algebras is an isomorphism of the corre-

sponding structures. Conversely,

x q y = 2 [(x + y)2 ’ x2 ’ y 2 ]

1

de¬nes on the underlying vector space J of a cubic norm structure J a Jordan

multiplication and an isomorphism of cubic norm structures is an isomorphism for

this multiplication.

520 IX. CUBIC JORDAN ALGEBRAS

38.A. The Springer Decomposition. Let L = F [x] be a cubic ´tale subal-

e

gebra of a Freudenthal algebra J (in the new sense). Since L is ´tale, the bilinear

e

trace form TJ |L is nonsingular and there is an orthogonal decomposition

V = L⊥ ‚ J 0 = { x ∈ J | TJ (x) = 0 }.

J =L⊥V with

We have TJ ( ) = T ( , 1) = TL/F ( ) and NJ ( ) = NL/F ( ) for ∈ L. It follows from

T ( 1 — 2 , v) = T ( 1 , 2 — v) = 0 for v ∈ V that — v ∈ V for ∈ L and v ∈ V . We

de¬ne

—¦ v = ’ — v,

so that —¦ v ∈ V for ∈ L and v ∈ V . Further, let Q : V ’ L and β : V ’ V be

the quadratic maps de¬ned by setting

v # = ’Q(v), β(v) ∈ L • V,

so that

( , v)# = #

’ Q(v), β(v) ’ —¦ v .

We have

SJ (v) = TJ (v # ) = ’TL/F Q(v)

since T β(v) = 0. Furthermore, putting β(v, w) = β(v + w) ’ β(v) ’ β(w), we get

v — w = ’bQ (v, w), β(v, w) .

(38.5) Example. Let J = H3 (C, 1) be a reduced Freudenthal algebra and let

L = F — F — F ‚ J be the set of diagonal elements. Then V is the space of

matrices

«

0 c 3 c2

¯

v = c 3 0 c 1 , c i ∈ C

¯

c 2 c1 0

¯

and the “—¦”-action of L on V is given by

« «

0 c3

¯ c2 0 »3 c 3

¯ »2 c 2

c 3 0 = »3 c3 »1 c 1

c1

¯ 0 ¯

(»1 , »2 , »3 ) —¦

c 2 c1

¯ 0 »2 c 2

¯ »1 c 1 0

Identifying C • C • C with V through the map

«

0 c3

¯ c2

v = (c1 , c2 , c3 ) ’ c3 0 c1

¯

c 2 c1

¯ 0

the action of L on V is diagonal, hence V is an L-module. We have

Q(v) = (c1 c1 , c2 c2 , c3 c3 )

for v = (c1 , c2 , c3 ), so that (V, Q) is a quadratic space over L. Furthermore we get

β(v) = (c2 c3 , c3 c1 , c1 c2 ),

hence β( —¦ v) = # —¦ β(v), Q β(v) = Q(v)# , and bQ v, β(v) = NJ (v) ∈ F . Thus

(V, L, Q, β) is a twisted composition.

The properties of the “—¦”-action described in Example (??) hold in general:

§38. CUBIC JORDAN ALGEBRAS 521

(38.6) Theorem. (1) Let (J, N, #, T, 1) be a cubic Freudenthal algebra, let L be a

cubic ´tale subalgebra of dimension 3 and let V = L⊥ for the bilinear trace form T .

e

The operation L—V ’ V given by ( , v) ’ —¦v de¬nes the structure of an L-module

on V such that (L, Q) is a quadratic space and (V, L, Q, β) is a twisted composition.

(2) For any twisted composition (V, L, Q, β), the cubic structure (J, N, #, T, 1) on

the vector space J(L, V ) = L • V given by

N ( , v) = NL/F ( ) + bQ v, β(v) ’ TL Q(v) ,

( , v)# = #

’ Q(v), β(v) ’ —¦ v ∈ L • V

T ( 1 , v1 ), ( 2 , v2 ) = TL/F ( 1 2) + TL/F bQ (v1 , v2 )

is a Freudenthal algebra. Furthermore we have

SJ ( , v) = TJ ( , v)# = TJ ( #

) ’ TJ Q(v) .

Proof : (??) It su¬ces to check that V is an L-module over a separable closure,

and there we may assume by (??) (which also holds for Freudenthal algebras in

the new sense) that L = F — F — F is diagonal in some H3 (C, ±). The claim

then follows from Example (??). Claim (??) can also be checked rationally, see

Petersson-Racine [?, Proposition 2.1] (or Springer [?], if char F = 2).

(??) If L is split and V = C • C • C, we may identify J(V, L) with H3 (C, 1)

as in Example (??). The general case then follows by descent.

We say that the Freudenthal algebra J(V, L) is the Springer construction asso-

ciated with the twisted composition (V, L, Q, β). This construction was introduced

by Springer for cyclic compositions of dimension 8, in relation to exceptional Jordan

algebras. Conversely, given L ‚ J ´tale of dimension 3, we get a Springer decom-

e

position J = L • V . Springer decompositions for arbitrary cubic structures were

¬rst considered by Petersson-Racine [?, Section 2]. Any Freudenthal algebra is (in

many ways) a Springer construction.

Let “s = (V, L, Q, β) be a split twisted composition of dimension 8. Its associ-

ated algebraic group of automorphisms is Spin8 S3 (see (??)). The corresponding

Freudenthal algebra Js = J(V, L) is split; we recall that by Theorem (??) its auto-

morphism group de¬nes a simple split algebraic group G of type F4 .

(38.7) Corollary. The map “s ’ Js induces an injective group homomorphism

S3 ’ G. The corresponding map in cohomology H 1 (F, Spin8 S3 ) ’

Spin8

H 1 (F, G), which associates the class of J(V, L) to a twisted composition (V, L, Q, β),

is surjective.

Proof : Let (V, L, Q, β) be a twisted split composition of dimension 8. Clearly any

automorphism of (V, L, Q, β) extends to an automorphism of J(L, V ) and conversely

any automorphism of J(L, V ) which maps L to L restricts to an automorphism of

(V, L, Q, β). This shows the injectivity of Spin8 S3 ’ G. The second claim follows

from the facts that H 1 (F, Spin8 S3 ) classi¬es twisted compositions of dimension 8

(Proposition (??)), that H 1 (F, G) classi¬es Albert algebras (Proposition (??)) and

that any Albert algebra admits a Springer decomposition.

(38.8) Theorem. Let J(V, L) be the Springer construction associated with a twis-

ted composition (V, L, Q, β). Then J(V, L) has zero divisors if and only if the

twisted composition (V, L, Q, β) is similar to a Hurwitz composition “(C, L) for

some Hurwitz algebra C. Furthermore, we have J(V, L) H3 (C, ±) for some ±

(and the same Hurwitz algebra C).

522 IX. CUBIC JORDAN ALGEBRAS

Proof : By Theorem (??) the composition (V, L, Q, β) is similar to a Hurwitz compo-

sition “(C, L) if and only if there exists v ∈ V such that β(v) = »v and N (v) = »#

for some » = 0 ∈ L; we then have (v, »)# = 0 in J(V, L). By Exercise ?? of this

chapter, this is equivalent with the existence of zero divisors in J(V, L) (see also the

proof of (??)). Hence Theorem (??) implies that J(V, L) is a reduced Freudenthal

algebra H3 (C , ±) for some Hurwitz algebra C . It remains to be shown that C C .

We consider the case where F has characteristic di¬erent from 2 (and leave the other

case as an exercise). For any bilinear form (x, y) ’ b(x, y) over L, let (TL/F )— (b)

be its transfer to F , i.e.,

(TL/F )— (b)(x, y) = TL/F b(x, y) .

The bilinear trace form of J(V, L) is the bilinear form:

T = (TL/F )— ( 1 L ) ⊥ (TL/F )— (bQ )

and bQ is extended from the bilinear form bQ0 = 2 L ⊥ δbNC over F (see Lemma

0

0

(??)). Let bNC = bC and bNC = bC . By Frobenius reciprocity (see Scharlau [?,

0

Theorem 5.6, p. 48]) we get

⊥ δb0 ).

T (TL/F )— ( 1 L ) ⊥ (TL/F )— ( 1 L ) — ( 2 L C

Since (TL/F )— ( 1 L ) = 1, 2, 2δ (see (??), (??)), it follows that

1, 2, 2δ ⊥ δb0 — 1, 2, 2δ ⊥ 2, 1, δ

T C

1, 2, 2 ⊥ bC — 2, δ, 2δ

1, 1, 1 ⊥ bC — 2, δ, 2δ .

Thus

bC — 2, δ, 2δ b C — ± 1 , ±2 , ±3 ,

since an isomorphism J(V, L) H3 (C , ±) implies that the corresponding trace

forms are isomorphic. The last claim then follows from Lemma (??) and Theo-

rem (??).

§39. The Tits Construction

Let K be a quadratic ´tale algebra with conjugation ι and let B be an associa-

e

tive separable algebra of degree 3 over K with a unitary involution „ (according to

an earlier convention, we also view K as a cubic separable K-algebra if char F = 3).

The generic norm NB of B de¬nes a cubic structure on B (as a K-algebra) and

restricts to a cubic structure on H(B, „ ). Let (u, ν) ∈ H(B, „ ) — K — be such that

NB (u) = ν„ (ν).

One can take for example (u, ν) = (1, 1). On the set

J(B, „, u, ν) = H(B, „ ) • B,

let 1 = (1, 0) and

N (a, b) = NB (a) + TK/F νNB (b) ’ TB abu„ (b)

(a, b)# = a# ’ bu„ (b), „ (ν)„ (b)# u’1 ’ ab

for (a, b) ∈ H(B, „ ) • B. Further let

T (a1 , b1 ), (a2 , b2 ) = TB (a1 a2 ) + TK/F TB b1 u„ (b2 ) .

§39. THE TITS CONSTRUCTION 523

(39.1) Theorem. The space J(B, „, u, ν) admits a Freudenthal cubic structure

with 1 as unit, N as norm, # as Freudenthal adjoint and T as bilinear trace form.

Furthermore we have SJ (a, b) = SB (a) ’ TB bu„ (b) for the quadratic trace SJ .

Reference: A characteristic-free proof is in Petersson-Racine [?, Theorem 3.4], see

also McCrimmon [?, Theorem 7]. The claim is also a consequence of Proposi-

tion (??) (see Corollary (??)) if char F = 3. The last claim follows from SJ (x) =

TJ (x# ) and TJ (x) = T (x, 1).

If char F = 2, the Jordan product of J(B, „, u, ν) is given by (see p. ??):

a1 b1

1

a2 2 (a1 a2 + a 2 a1 ) (a2 b1 )—

1

„ (u) „ (b1 ) — „ (b2 ) ν ’1

b2 (a1 b2 )— b1 ν„ (b2 ) + b2 ν„ (b1 ) + 2 —

where x = 1 TrdB (x) ’ x and x— denotes x as an element of the second com-

2

ponent B. The cubic structure J(B, „, u, ν) described in Theorem (??) is a Tits

construction or a Tits process and the pair (u, ν) is called an admissible pair for

(B, „ ). The following lemma describe some useful allowed changes for admissible

pairs.

(39.2) Lemma. (1) Let (u, v) be an admissible pair for (B, „ ). For any w ∈

B — , wu„ (w), νNB (w) is an admissible pair for (B, „ ) and (a, b) ’ (a, bw) is an

isomorphism

∼

J(B, „, u, ν) ’ J B, „, wu„ (w), νNB (w) .

’

(2) For any Tits construction J(B, „, u, ν), there is an isomorphic Tits construction

J(B, „, u , ν ) with NB (u ) = 1 = ν „ (ν ).

Proof : The ¬rst claim reduces to a tedious computation, which we leave as an

exercise (see also Theorem (??)). For the second, we take w = ν ’1 u in (??).

An exceptional Jordan algebra of dimension 27 of the form J(B, „, u, ν) where

B is a central simple algebra over a quadratic ¬eld extension K of F , is classically

called a second Tits construction. The case where K is not a ¬eld also has to be

considered. Let J(B, „, u, ν) be a Tits process with K = F — F , B = A — Aop where

A is either central simple or cubic ´tale over F and „ is the exchange involution.

e

By Lemma (??) we may assume that the admissible pair (u, ν) is of the form

1, (», »’1 ) , » ∈ F — . Projecting B onto the ¬rst factor A induces an isomorphism

of vector spaces

J(B, „, u, ν) A • A • A,

the norm is given by NJ (a, b, c) = NA (a) + »NA (b) + »’1 NA (c) ’ TA (abc) and the

Freudenthal adjoint on A • A • A reduces to

(a, b, c)# = (a# ’ bc, »’1 c# ’ ab, »b# ’ ca)

where a ’ a# is the Freudenthal adjoint of A+ (which is a cubic algebra!); thus

we have SJ (a, b, c) = SA (a) ’ TA (bc) and the bilinear trace form is given by

TJ (a1 , b1 , c1 ), (a2 , b2 , c2 ) = TA (a1 b1 ) + TA (a2 b3 ) + TA (a3 b2 ).

524 IX. CUBIC JORDAN ALGEBRAS

If char F = 2, the Jordan product is

(a1 , b1 , c1 ) q (a2 , b2 , c2 ) =

a1 q a2 + b1 c2 + b2 c1 , a1 b2 + a2 b1 + (2»)’1 (c1 — c2 ), c2 a1 + c1 a2 + 2 »(b1 — b2 ) ,

1

¯ ¯

where

a = 2 a — 1 = 1 TA (a) · 1 ’ 2 a.

¯1 1

2

Conversely we can associate to a pair (A, »), A central simple of degree 3 or cubic

´tale over F and » ∈ F — , a Freudenthal algebra J(A, ») = A • A • A, with norm,

e

Freudenthal product and trace as given above. The algebra J(A, ») is (classically)

a ¬rst Tits construction if A is central simple. Any ¬rst Tits construction J(A, »)

extends to a Tits process J A — Aop , σ, 1, (», »’1 ) over F — F . According to the

classical de¬nitions, we shall say that J(B, „, u, ν) is a second Tits process if K is

a ¬eld and that J(A, ») is a ¬rst Tits process.

(39.3) Remark. (See [?, p. 308].) Let (A, ») = A • A • A be a ¬rst Tits process.

To distinguish the three copies of A in J(A, »), we write

J(A, ») = A+ • A1 • A2

and denote a ∈ A as a, a1 , resp. a2 if we consider it as an element of A+ , A1 , or A2 .

The ¬rst copy admits the structure of an associative algebra, A1 (resp. A2 ) can be

characterized by the fact that it is a subspace of (A+ )⊥ (for the bilinear trace form)

such that a q a1 = ’a — a1 (resp. a2 q a = ’a — a2 ) de¬nes the structure of a left

A-module on A1 (resp. right A-module on A2 ).

(39.4) Proposition. For any second Tits process J(B, „, u, ν) over F , B a K-

algebra, J(B, „, u, ν) — K is isomorphic to the ¬rst Tits process J(B, ν) over K.

Conversely, any second Tits process J(B, „, u, ν) over F is the Galois descent of

the ¬rst Tits process (B, ν) over K under the ι-semilinear automorphism

(a, b, c) ’ „ (a), „ (c)u’1 , u„ (b) .

Proof : An isomorphism

∼

J(B, „, u, ν) — K ’ J(B, ν)

’

is induced by (a, b) ’ a, b, u„ (b) . The last claim follows by straightforward

computations.

(39.5) Examples. Assume that char F = 3.

(1) Any cubic ´tale F -algebra L can be viewed as a Tits construction over F ; if

e

√

3

L = F ( »), then L is isomorphic to the ¬rst Tits construction (F, »). In general

there exist a quadratic ´tale F -algebra K and some element ν ∈ K with NK (ν) = 1

e√

such that L — K K( 3 ν) and L is the second Tits construction (K, ι, 1, ν) (see

Proposition (??)).

(2) Let A be central simple of degree 3 over F . We write A as a crossed product

A = L • Lz • Lz 2 with L cyclic and z 3 = » ∈ F — , z = ρ( )z and ρ a generator of

Gal(L/F ); the map A ’ L • L • L given by a + bz + cz 2 ’ a, ρ(b), »ρ2 (c) is an

isomorphism of A+ with the ¬rst Tits construction (L, »).

§39. THE TITS CONSTRUCTION 525

(3) Let (B, „ ) be central simple of degree 3 with a distinguished unitary involution

„ over K. In view of Proposition (??) and Corollary (??), there exists a cubic ´tale

e

F -algebra L with discriminant ∆(L) K such that

B = L — K • (L — K)z • (L — K)z 2 z3 = » ∈ F —,

with „ (z) = z.

The K-algebra L — K is cyclic over K; let ρ ∈ Gal(L — K/K) be such that zξz ’1 =

ρ(ξ) for ξ ∈ L — K. We have

L1 = { ξ ∈ L — K | ρ —¦ (1 — ι)(ξ) = ξ } L,

L2 = { ξ ∈ L — K | ρ2 —¦ (1 — ι)(ξ) = ξ } L

and (1 — ι)(L1 ) = L2 , so that

H(B, „ ) = L • L1 • L2 L•L•L

and a check shows this is an isomorphism of H(B, „ ) with the ¬rst Tits construction

(L, »). Since the exchange involution on A — Aop is distinguished, we see that

H(B, „ ) is a ¬rst Tits construction if and only if „ is distinguished, if and only if

SB |H(B,„ )0 has Witt index at least 3 (see Proposition (??) for the last equivalence).

(4) Let (B, „ ) be central simple with a unitary involution over K and assume that

H(B, „ ) contains a cyclic ´tale algebra L over F . By Albert [?, Theorem 1] we may

e

write B as a crossed product

B = L — K • (L — K)z • (L — K)z 2

(39.6)

with z 3 = ν ∈ K — such that NK (ν) = 1; furthermore the involution „ is determined

by „ (z) = uz ’1 with u ∈ L such that NB (u) = 1. In this case H(B, „ ) is isomorphic

to the second Tits process (L — K, 1 — ιK , u, ν).

(5) A Tits construction J = J(L — K, 1 — ιK , u, ν) with L cubic ´tale is of di-

e

mension 9, hence by Theorem (??) it is of the form H(B, „ ) for a central simple

algebra B of degree 3 over an ´tale quadratic F -algebra K1 and a unitary involution

e

„ . We may describe (B, „ ) more explicitly: if L is cyclic, K1 = K, and B is as

in (??). If L is not cyclic, we replace L by L2 = L — ∆(L), where ∆(L) is the

discriminant of L, and obtain (B2 , „2 ) over ∆(L) from (??). Let φ be the descent

φ

on B2 given by φ = 1 — ι∆(L) — ιK on L — ∆(L) — K and φ(z) = z ’1 . Then B = B2

(A — Aop , exchange) if and

and K1 = ∆(L) — K. In particular we have (B, „ )

only if ∆(L) K.

(6) Let J be a Freudenthal algebra of dimension 9 over F and let L be a cubic ´tale

e

subalgebra of J. We may describe J as a Tits construction J(L — K, 1 — ιK , u, ν)

as follows. Let J = L • V be the Springer decomposition induced by L. Then V is

a twisted composition (V, L, Q, β) and V is of dimension 2 over L; by Proposition

(??) V admits the structure of a hermitian L — K-space for some quadratic ´tale e

—

F -algebra K. Let V = (L — K)v; let u = Q(v) ∈ L and let β(v) = xv, x ∈ L — K.

It follows from bQ v, β(v) ∈ F that (x + x)u ∈ F , where x ’ x is the extension of

the conjugation ιK of K to L — K. Similarly Q β(v) = u# implies that xxu2 =

NL/F (u) ∈ F — . Both imply that xu (or xu) lies in K and NK/F (xu) = nL/F (u).

Let J be the Tits construction J(L — K, 1 — ιK , u, xu). The map J ’ J given by

(a, b) ’ (a, bv) is an isomorphism of Jordan algebras.

(7) A ¬rst Tits construction J(A, 1) with A cubic ´tale or central simple of de-

e

gree 3 is always a split Freudenthal algebra: this is clear for cubic ´tale algebras

e

by Example (??). So let A be central simple. Taking a ∈ A such that a3 ∈ F — ,

526 IX. CUBIC JORDAN ALGEBRAS

we see that a# = a2 , so that (a, a, a)# = 0 and, by Exercise ?? of this chapter

J(A, 1) is reduced. Theorem (??) then implies that J(A, 1) H3 (C, ±). Let L1

be a cubic extension which splits A. By Theorem (??) C — L1 is split, hence by