σ (xij )1¤i,j¤3 =

θ(x23 ) θ(x13 ) θ(x33 )

and the linear form f by

«

x11 s12 x13

s21 x23 = TrdQ (x11 ) + f1 (s33 ),

θ(x11 )

f

θ(x23 ) θ(x13 ) s33

for x11 , x13 , x23 ∈ Q and s12 , s21 , s33 ∈ Sym(Q, θ).

It is readily veri¬ed that

±«

x1 x2 x3

I = 0 0

0 x1 , x 2 , x 3 ∈ Q

0 0 0

is an isotropic right ideal, and that disc(A, σ, f ) = disc(Q, θ, f1 ), hence Z(A, σ, f )

Z.

Let (B, „ ) = C(A, σ, f ), σ . Since rdim I = 2, we have 2 ∈ ind(A, σ, f ),

hence Corollary (??) yields 1 ∈ ind(B, „ ). This relation shows that B is split,

hence B M4 (Z), and „ is the adjoint involution with respect to an isotropic

4-dimensional hermitian form h over Z. Multiplying h by a suitable scalar, we may

assume that h has a diagonalization 1, ’1, 1, ’a for some a ∈ F — . Corollary (??)

then shows that D(B, „ ) is Brauer-equivalent to the quaternion algebra (Z, a) F .

Since D(B, „ ) A, we have (Z, d)F (Z, a)F , hence a ≡ d mod N (Z/F ) and

therefore

h 1, ’1, 1, ’d .

The same arguments apply to every isotropic quadratic pair (σ, f ) on A such

that Z(A, σ, f ) Z: for every such quadratic pair, we have C(A, σ, f ), σ

M4 (Z), σh where h 1, ’1, 1, ’d , hence also

(A, σ, f ) D M4 (Z), σh , σh , fD .

This proves uniqueness of the quadratic pair (σ, f ) up to conjugation.

232 IV. ALGEBRAS OF DEGREE FOUR

Indices. Let (B, „ ) ∈ A3 and (A, σ, f ) ∈ D3 correspond to each other un-

der the equivalence A3 ≡ D3 . Let K be the center of B, which is isomorphic to

F ( disc(σ, f )) if char F = 2 and to F „˜’1 disc(σ, f ) if char F = 2. Our goal

is to relate the indices ind(A, σ, f ) and ind(B, „ ). For clarity, we consider the case

where K F — F separately.

(15.38) Proposition. Suppose K F — F , hence (B, „ ) (E — E op , µ) for some

central simple F -algebra E, where µ is the exchange involution. The only possibili-

ties for ind(A, σ, f ) are

{0}, {0, 1} and {0, 1, 2, 3}.

Moreover,

ind(A, σ, f ) = {0} ⇐’ ind(B, „ ) = {0} ⇐’ ind E = 4,

ind(A, σ, f ) = {0, 1} ⇐’ ind(B, „ ) = {0, 2} ⇐’ ind E = 2,

ind(A, σ, f ) = {0, 1, 2, 3} ⇐’ ind(B, „ ) = {0, 1, 2} ⇐’ ind E = 1.

Proof : Since deg A = 6, we have ind(A, σ, f ) ‚ {0, 1, 2, 3}. If 3 ∈ ind(A, σ, f ), then

A splits since ind A is a power of 2 which divides all the integers in ind(A, σ, f ).

In that case, we have (A, σ, f ) EndF (V ), σq , fq for some hyperbolic quadratic

space (V, q), hence ind(A, σ, f ) = {0, 1, 2, 3}.

Since K F — F , Proposition (??) shows that ind(A, σ, f ) = {0, 2}. There-

fore, if 2 ∈ ind(A, σ, f ), we must also have 1 or 3 ∈ ind(A, σ, f ), hence, as above,

(A, σ, f ) EndF (V ), σq , fq for some quadratic space (V, q) with w(V, q) ≥ 2.

Since disc(σ, f ) = disc q is trivial, the inequality w(V, q) ≥ 2 implies q is hyperbolic,

hence ind(A, σ, f ) = {0, 1, 2, 3}. Therefore, the only possibilities for ind(A, σ, f ) are

those listed above.

The relations between ind(B, „ ) and ind E readily follow from the de¬nition of

ind(E — E op , µ), and the equivalences ind(A, σ, f ) = {0, 1} ⇐’ ind E = 2 and

ind(A, σ, f ) = {0, 1, 2, 3} ⇐’ ind E = 1 follow from (??) and (??).

(15.39) Proposition. Suppose K is a ¬eld. The only possibilities for ind(A, σ, f )

are

{0}, {0, 1}, {0, 2} and {0, 1, 2}.

Moreover,

ind(A, σ, f ) = {0} ⇐’ ind(B, „ ) = {0},

ind(A, σ, f ) = {0, 1} ⇐’ ind(B, „ ) = {0, 2},

ind(A, σ, f ) = {0, 2} ⇐’ ind(B, „ ) = {0, 1},

ind(A, σ, f ) = {0, 1, 2} ⇐’ ind(B, „ ) = {0, 1, 2}.

Proof : If 3 ∈ ind(A, σ, f ), then (σ, f ) is hyperbolic, hence its discriminant is trivial,

by (??). This contradicts the hypothesis that K is a ¬eld. Therefore, we have

ind(A, σ, f ) ‚ {0, 1, 2}.

To prove the correspondence between ind(A, σ, f ) and ind(B, „ ), it now su¬ces

to show that 1 ∈ ind(A, σ, f ) if and only if 2 ∈ ind(B, „ ) and that 2 ∈ ind(A, σ, f )

if and only if 1 ∈ ind(B, „ ).

§16. BIQUATERNION ALGEBRAS 233

If 1 ∈ ind(A, σ, f ), then A is split, and Proposition (??) shows that 2 ∈

ind(B, „ ). Conversely, if 2 ∈ ind(B, „ ), then B is Brauer-equivalent to a quater-

nion algebra and „ is hyperbolic. It follows from (??) that 1 ∈ ind(A, σ, f ) in that

case. If 2 ∈ ind(A, σ, f ), then Corollary (??) yields 1 ∈ ind(B, „ ). Conversely, if

1 ∈ ind(B, „ ), then B splits and „ is the adjoint involution with respect to some

isotropic 4-dimensional hermitian form. By (??) (if „ is not hyperbolic) or (??) (if

„ is hyperbolic), it follows that 2 ∈ ind(A, σ, f ).

(15.40) Remark. The correspondence between ind(A, σ, f ) and ind(B, „ ) may be

summarized in the following relations (which hold when K F — F as well as when

K is a ¬eld):

1 ∈ ind(A, σ, f ) ⇐’ 2 ∈ ind(B, „ ), 2 ∈ ind(A, σ, f ) ⇐’ 1 ∈ ind(B, „ ),

3 ∈ ind(A, σ, f ) ⇐’ ind(A, σ, f ) = {0, 1, 2, 3}

⇐’ ind(B, „ ) = {0, 1, 2} and K F —F .

§16. Biquaternion Algebras

Algebras which are tensor products of two quaternion algebras are called bi-

quaternion algebras. Such algebras are central simple of degree 4 and exponent 2

(or 1). Albert proved the converse:

(16.1) Theorem (Albert [?, p. 369]). Every central simple algebra of degree 4 and

exponent 2 is a biquaternion algebra.

We present three proofs. The ¬rst two proofs rely heavily on the results of §??,

whereas the third proof, due to Racine, is more self-contained.

Throughout this section, A is a central simple algebra of degree 4 and exponent

1 or 2 over an arbitrary ¬eld F .

C ± (V, q) for some 6-dimensional

First proof (based on A3 ≡ D3 ): By (??), A

quadratic space (V, q) of trivial discriminant. The result follows from the structure

of Cli¬ord algebras of quadratic spaces.

Explicitly, if char F = 2 we may assume (after a suitable scaling) that q has a

diagonalization of the form

q = a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2

for some a1 , b1 , a2 , b2 ∈ F — . Let (e1 , . . . , e6 ) be an orthogonal basis of V which

yields that diagonalization. The even Cli¬ord algebra has a decomposition

C0 (V, q) = Q1 —F Q2 —F Z

where Q1 is the F -subalgebra generated by e1 ·e2 and e1 ·e3 , Q2 is the F -subalgebra

generated by e4 · e5 and e4 · e6 , and Z = F · 1 • F · e1 · e2 · e3 · e4 · e5 · e6 is the

F — F , hence C + (V, q) C ’ (V, q)

center of C0 (V, q). We have Z Q 1 — Q2 .

’1 ’1

Moreover, (1, b1 e2 · e3 , a1 e1 · e3 , e1 · e2 ) is a quaternion basis of Q1 which shows

Q1 (a1 , b1 )F , and (1, b’1 e5 · e6 , a’1 e4 · e6 , e5 · e6 ) is a quaternion basis of Q2 which

2 2

shows Q2 (a2 , b2 )F . Therefore, C ± (V, q) is a biquaternion algebra.

Similar arguments hold when char F = 2. We may then assume

q = [1, a1 b1 + a2 b2 ] ⊥ [a1 , b1 ] ⊥ [a2 , b2 ]

for some a1 , b1 , a2 , b2 ∈ F . Let (e1 , . . . , e6 ) be a basis of V which yields that

decomposition. In C0 (V, q), the elements e1 · e3 and e1 · e4 (resp. e1 · e5 and e1 · e6 )

234 IV. ALGEBRAS OF DEGREE FOUR

generate a quaternion F -algebra Q1 a 1 , b1 (resp. Q2 a 2 , b2 F ). We have

F

a decomposition

C0 (V, q) = Q1 —F Q2 —F Z

where Z = F · 1 • F · (e1 · e2 + e3 · e4 + e5 · e6 ) is the center of C0 (V, q). Since

Z F — F , it follows that

C + (V, q) C ’ (V, q) Q 1 —F Q2 a 1 , b1 — a 2 , b2 .

F F

Second proof (based on B2 ≡ C2 ): By (??) and (??), the algebra A carries an in-

volution σ of symplectic type. In the notation of §??, we have (A, σ) ∈ C2 . The

proof of the equivalence B2 ≡ C2 in (??) shows that A is isomorphic to the even

Cli¬ord algebra of some nonsingular 5-dimensional quadratic form:

C0 Symd(A, σ)0 , sσ .

(A, σ)

The result follows from the fact that even Cli¬ord algebras of odd-dimensional

quadratic spaces are tensor products of quaternion algebras (Scharlau [?, Theo-

rem 9.2.10]).

Third proof (Racine [?]): If A is not a division algebra, the theorem readily follows

from Wedderburn™s theorem (??), which yields a decomposition:

A M2 (F ) —F Q

for some quaternion algebra Q. We may thus assume that A is a division algebra.

Our ¬rst aim is to ¬nd in A a separable quadratic extension K of F . By (??),

A carries an involution σ. If char F = 2, we may start with any nonzero element

u ∈ Skew(A, σ); then u2 ∈ Sym(A, σ), hence F (u2 ) F (u). Since [F (u) : F ] = 4

or 2, we get [F (u ) F ] = 2 or 1 respectively. We choose K = F (u2 ) in the

2:

¬rst case and K = F (u) in the second case. In arbitrary characteristic, one may

choose a symplectic involution σ on A and take for K any proper extension of F

in Symd(A, σ) which is not contained in Symd(A, σ)0 , by (??).

The theorem then follows from the following proposition, which also holds when

A is not a division algebra. Recall that for every ´tale quadratic F -algebra K with

e

—

nontrivial automorphism ι and for every a ∈ F , the symbol (K, a)F stays for the

quaternion F -algebra K • Kz where multiplication is de¬ned by zx = ι(x)z for

x ∈ K and z 2 = a.

(16.2) Proposition. Suppose K is an ´tale quadratic F -algebra contained in a

e

central simple F -algebra A of degree 4 and exponent 2. There exist an a ∈ F — and

a quaternion F -algebra Q such that

A (K, a)F — Q.

Proof : If K is not a ¬eld, then A is not a division algebra, hence Wedderburn™s

theorem (??) yields

A (K, 1)F — Q

for some quaternion F -algebra Q.

If K is a ¬eld, the nontrivial automorphism ι extends to an involution „ on A

by (??). The restriction of „ to the centralizer B of K in A is an involution of

§16. BIQUATERNION ALGEBRAS 235

the second kind. Since B is a quaternion algebra over K, Proposition (??) yields a

quaternion F -algebra Q ‚ B such that

B = Q —F K.

By (??), there is a decomposition

B = Q — F CB Q

where CB Q is the centralizer of Q in B. This centralizer is a quaternion algebra

which contains K, hence

CB Q (K, a)F

for some a ∈ F — . We thus get the required decomposition.

16.A. Albert forms. Let A be a biquaternion algebra over a ¬eld F of ar-

bitrary characteristic. The algebra »2 A is split of degree 6 and carries a canonical

quadratic pair (γ, f ) of trivial discriminant (see (??)). Therefore, there are quad-

ratic spaces (V, q) of dimension 6 and trivial discriminant such that

(»2 A, γ, f ) EndF (V ), σq , fq

where (σq , fq ) is the quadratic pair associated with q.

(16.3) Proposition. For a biquaternion algebra A and a 6-dimensional quadratic

space (V, q) of discriminant 1, the following conditions are equivalent:

(1) (»2 A, γ, f ) EndF (V ), σq , fq ;

(2) A — A C0 (V, q);

(3) M2 (A) C(V, q).

Moreover, if (V, q) and (V , q ) are 6-dimensional quadratic spaces of discriminant 1

which satisfy these conditions for a given biquaternion algebra A, then (V, q) and

» · q for some » ∈ F — .

(V , q ) are similar, i.e., q

The quadratic forms which satisfy the conditions of this proposition are called

Albert forms of the biquaternion algebra A (and the quadratic space (V, q) is called

an Albert quadratic space of A). As the proposition shows, an Albert form is

determined only up to similarity by A. By contrast, it is clear from condition (??)

or (??) that any quadratic form of dimension 6 and discriminant 1 is an Albert

form for some biquaternion algebra A, uniquely determined up to isomorphism.

Proof : (??) ’ (??) Condition (??) implies that C(»2 A, γ, f ) C0 (V, q). Since

2 op op

(??) shows that C(» A, γ, f ) A — A and since A A , we get (??).

(??) ’ (??) Since the canonical involution σq on C0 (V, q) = C EndF (V ), σq , fq

is of the second kind, we derive from (??):

(A — Aop , µ),

C EndF (V ), σq , fq , σq

where µ is the exchange involution. By comparing the discriminant algebras of both

sides, we obtain

(»2 A, γ, f ).

D C EndF (V ), σq , fq , σq , σq , fD

Corollary (??) (or Theorem (??)) shows that there is a natural transformation

D —¦ C ∼ IdD3 , hence

=

D C EndF (V ), σq , fq , σq , σq , fD EndF (V ), σq , fq

236 IV. ALGEBRAS OF DEGREE FOUR

and we get (??).

(??) ” (??) This follows from the structure of Cli¬ord algebras of quadratic

forms: see for instance Lam [?, Ch. 5, Theorem 2.5] if char F = 2; similar arguments

hold in characteristic 2.

Finally, if (V, q) and (V , q ) both satisfy (??), then

EndF (V ), σq , fq EndF (V ), σq , fq ,

hence (V, q) and (V , q ) are similar, by (??).

(16.4) Example. Suppose char F = 2. For any a1 , b1 , a2 , b2 ∈ F — , the quadratic

form

q = a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2

is an Albert form of the biquaternion algebra (a1 , b1 )F — (a2 , b2 )F . This follows

from the computation of the Cli¬ord algebra C(q). (See the ¬rst proof of (??); see

also (??) below.)

Similarly, if char F = 2, then for any a1 , b1 , a2 , b2 ∈ F , the quadratic form

[1, a1 b1 + a2 b2 ] ⊥ [a1 , b1 ] ⊥ [a2 , b2 ]

is an Albert form of the biquaternion algebra a1 , b1 — a 2 , b2 F, and, for a1 ,

F

a2 ∈ F , b1 , b2 ∈ F — , the quadratic form

[1, a1 + a2 ] ⊥ b1 [1, a1 ] ⊥ b2 [1, a2 ]

is an Albert form of [a1 , b1 )F — [a2 , b2 )F .

Albert™s purpose in associating a quadratic form to a biquaternion algebra

was to obtain a necessary and su¬cient quadratic form theoretic criterion for the

biquaternion algebra to be a division algebra.

(16.5) Theorem (Albert [?]). Let A be a biquaternion algebra and let q be an

Albert form of A. The (Schur ) index of A, ind A, and the Witt index w(q) are

related as follows:

ind A = 4 w(q) = 0;

if and only if

(in other words, A is a division algebra if and only if q is anisotropic);

ind A = 2 w(q) = 1;

if and only if

ind A = 1 w(q) = 3;

if and only if

(in other words, A is split if and only if q is hyperbolic).

Proof : This is a particular case of (??).

Another relation between biquaternion algebras and their Albert forms is the

following:

(16.6) Proposition. The multipliers of similitudes of an Albert form q of a bi-

quaternion algebra A are given by

G(q) = F —2 · NrdA (A— )

and the spinor norms by

Sn(q) = { » ∈ F — | »2 ∈ NrdA (A— ) }.

Proof : This is a direct application of (??).

§16. BIQUATERNION ALGEBRAS 237

Even though there is no canonical choice for an Albert quadratic space of a

biquaternion algebra A, when an involution of the ¬rst kind σ on A is ¬xed, an

Albert form may be de¬ned on the vector space Symd(A, σ) of symmetric elements if

σ is symplectic and on the vector space Skew(A, σ) if σ is orthogonal and char F = 2.

Moreover, Albert forms may be used to de¬ne an invariant of symplectic involutions,

as we now show.

16.B. Albert forms and symplectic involutions. Let σ be a symplectic

involution on the biquaternion F -algebra A. Recall from §?? (see (??)) that the

reduced characteristic polynomial of every symmetrized element is a square:

2

PrdA,s (X) = Prpσ,s (X)2 = X 2 ’ Trpσ (s)X + Nrpσ (s) for s ∈ Symd(A, σ).

Since deg A = 4, the polynomial map Nrpσ : Symd(A, σ) ’ F has degree 2. We

show in (??) below that Symd(A, σ), Nrpσ is an Albert quadratic space of A.

A key tool in this proof is the linear endomorphism of Symd(A, σ) de¬ned by

x = Trpσ (x) ’ x for x ∈ Symd(A, σ).

Since Prpσ,x (x) = 0 for all x ∈ Symd(A, σ), we have

Nrpσ (x) = xx = xx for x ∈ Symd(A, σ).

(16.7) Lemma. For x ∈ Symd(A, σ) and a ∈ A— ,

axσ(a) = NrdA (a)σ(a)’1 xa’1 .

Proof : Since both sides of the equality above are linear in x, it su¬ces to show that

the equality holds for x in some basis of Symd(A, σ). It is readily seen by scalar

extension to a splitting ¬eld that Symd(A, σ) is spanned by invertible elements.

Therefore, it su¬ces to prove the equality for invertible x. In that case, the property

follows by comparing the equalities

NrdA (a) Nrpσ (x) = Nrpσ axσ(a) = axσ(a) · axσ(a)

and

Nrpσ (x) = axσ(a) · σ(a)’1 xa’1 .

(16.8) Proposition. The quadratic space Symd(A, σ), Nrpσ is an Albert quad-

ratic space of A.

Proof : For x ∈ Symd(A, σ), let i(x) = x x ∈ M2 (A). Since xx = xx = Nrpσ (x),

0

0

we have i(x)2 = Nrpσ (x), hence the universal property of Cli¬ord algebras shows

that i induces an F -algebra homomorphism

(16.9) i— : C Sym(A, σ), Nrpσ ’ M2 (A).

This homomorphism is injective since C Sym(A, σ), Nrpσ is simple, and it is sur-

jective by dimension count.

(16.10) Remark. Proposition (??) shows that A contains a right ideal of reduced

dimension 2 if and only if the quadratic form Nrpσ is isotropic. Therefore, A is a

division algebra if and only if Nrpσ is anisotropic; in view of (??), this observation

yields an alternate proof of a (substantial) part of Albert™s theorem (??).

238 IV. ALGEBRAS OF DEGREE FOUR

The isomorphism i— of (??) may also be used to give an explicit description of

the similitudes of the Albert quadratic space Symd(A, σ), Nrpσ , thus yielding an

alternative proof of the relation between Cli¬ord groups and symplectic similitudes

in (??).

(16.11) Proposition. The proper similitudes of Symd(A, σ), Nrpσ are of the

form

x ’ »’1 axσ(a)

where » ∈ F — and a ∈ A— .

The improper similitudes of Symd(A, σ), Nrpσ are of the form

x ’ »’1 axσ(a)

where » ∈ F — and a ∈ A— . The multiplier of these similitudes is »’2 NrdA (a).

Proof : Since Nrpσ (x) = Nrpσ (x) for all x ∈ Sym(A, σ), it follows from (??) that

the maps above are similitudes with multiplier »’2 NrdA (a) for all » ∈ F — and

a ∈ A— .

Conversely, let f ∈ GO Symd(A, σ), Nrpσ be a similitude and let ± = µ(f )

be its multiplier. The universal property of Cli¬ord algebras shows that there is an

isomorphism

f— : C Symd(A, σ), Nrpσ ’ M2 (A)

0 ±’1 f (x)

which maps x ∈ Symd(A, σ) to . By comparing f— with the isomor-

f (x) 0

phism i— of (??), we get an automorphism f— —¦i’1 of M2 (A). Note that the checker-

—

board grading of M2 (A) corresponds to the canonical Cli¬ord algebra grading under

both f— and i— . Therefore, f— —¦ i’1 is a graded automorphism, and f— —¦ i’1 = Int(u)

— —

for some u ∈ GL2 (A) of the form

v0 0 v

u= or .

0w w 0

Moreover, inspection shows that the automorphism C(f ) of C0 Symd(A, σ), Nrpσ

induced by f ¬ts in the commutative diagram

C(f )

C0 (Nrpσ ) ’ ’ ’ C0 (Nrpσ )

’’

¦ ¦

¦ ¦i

i— —

f— —¦i’1

’ ’—’

A—A ’’ A—A

where we view A—A as A A ‚ M2 (A). Therefore, in view of (??), the similitude f

0

0

is proper if and only if f— —¦ i’1 maps each component of A — A into itself. This

—

means that f is proper if u = v w and improper if u = w v .

0 0

0 0

’1

In particular, if f (x) = » axσ(a) for x ∈ Symd(A, σ), then

» NrdA (a)’1 axσ(a)

0

f— —¦ i’1 i(x) =

— ’1

» axσ(a) 0

and (??) shows that the right side is

»σ(a)’1 »’1 σ(a) »σ(a)’1

0 0x 0 0

· · = Int i(x) .

’1

0 a x0 0 a 0 a

§16. BIQUATERNION ALGEBRAS 239

Since the matrices of the form i(x), x ∈ Symd(A, σ) generate M2 (A) (as Symd(A, σ)

generates C Symd(A, σ), Nrpσ ), it follows that

»σ(a)’1 0

i’1