qσ (ei )

hence pσ (ei ) = »pσ (ei ) for i = 1, . . . , 6. By linearity of pσ and pσ , it follows that

pσ (x) = »pσ (x) for all x ∈ Skew(A, σ).

(16.23) Proposition. Let pσ be a non-zero linear endomorphism of Skew(A, σ)

such that xpσ (x) ∈ F for all x ∈ Skew(A, σ), and let

qσ (x) = xpσ (x) ∈ F for x ∈ Skew(A, σ).

The quadratic form qσ is nonsingular; for x ∈ Skew(A, σ) we have qσ (x) = pσ (x)x,

and qσ (x) = 0 if and only if x ∈ A— . Moreover, Skew(A, σ), qσ is an Albert

quadratic space of A.

Proof : Proposition (??) shows that pσ is a scalar multiple of an endomorphism

satisfying (??.??) and (??.??); therefore, pσ also satis¬es these conditions. It fol-

lows that qσ (x) = pσ (x)x for all x ∈ Skew(A, σ) and qσ (x) = 0 if and only if x is

invertible.

In order to show that qσ is nonsingular, we again consider a decomposition of A

into a tensor product of two quaternion subalgebras, so

A = Q 1 —F Q2

and set θ = γ1 — γ2 , which is the tensor product of the canonical involutions on Q1

and Q2 . As observed in the proof of (??), we have

Skew(A, σ) = (Q0 — 1) • (1 — Q0 ),

1 2

and we may consider the endomorphism pθ of Skew(A, σ) de¬ned by

pθ (x1 — 1 + 1 — x2 ) = x1 — 1 ’ 1 — x2

for x1 ∈ Q0 and x2 ∈ Q0 . Denoting qθ = xpθ (x) for x ∈ Skew(A, σ), we then have

1 2

qθ (x1 — 1 + 1 — x2 ) = x2 ’ x2 = ’ NrdQ1 (x1 ) + NrdQ2 (x2 ),

1 2

hence qθ is a nonsingular quadratic form.

If v ∈ Sym(A, θ) © A— is such that σ = Int(v) —¦ θ, (??) shows that there exists

» ∈ F — such that pσ (x) = »vpθ (xv) for all x ∈ Skew(A, σ). Then qσ (x) = »qθ (xv),

hence multiplication on the right by v de¬nes a similitude

∼

Skew(A, σ), qσ ’ Skew(A, θ), qθ

’

with multiplier ». Since qθ is nonsingular, it follows that qσ is also nonsingular.

248 IV. ALGEBRAS OF DEGREE FOUR

To complete the proof, consider the map

i : Skew(A, σ) ’ M2 (A)

de¬ned by

0 pσ (x)

i(x) = for x ∈ Skew(A, σ).

x 0

We have i(x)2 = qσ (x) for all x ∈ Skew(A, σ), hence the universal property of

Cli¬ord algebras shows that i induces an F -algebra homomorphism

i— : C Skew(A, σ), qσ ’ M2 (A).

(Compare with (??).) Since qσ is nonsingular, it follows that the Cli¬ord algebra

C Skew(A, σ), qσ is simple, hence i— is injective. It is also surjective by dimension

count, hence (??) shows that Skew(A, σ), qσ is an Albert quadratic space of A.

(16.24) Example. Let A = (a1 , b1 )F — (a2 , b2 )F . If θ = γ1 — γ2 is the tensor prod-

uct of the canonical involutions on the quaternion algebras (a1 , b1 )F and (a2 , b2 )F ,

the computations in the proof of (??) show that one can take

and qθ (x1 — 1 + 1 — x2 ) = x2 ’ x2

pθ (x1 — 1 + 1 — x2 ) = x1 — 1 ’ 1 — x2 1 2

for xi ∈ (ai , bi )0 , i = 1, 2. Therefore, qθ has the following diagonalization:

F

qθ = a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2 .

(Compare with (??).)

We now list a few properties of the endomorphism pσ de¬ned in (??).

(16.25) Proposition. Let pσ be a non-zero linear endomorphism of Skew(A, σ)

such that xpσ (x) ∈ F for all x ∈ Skew(A, σ), and let qσ : Skew(A, σ) ’ F be the

(Albert) quadratic map de¬ned by

qσ (x) = xpσ (x) for x ∈ Skew(A, σ).

(1) For all a ∈ A— and x ∈ Skew(A, σ), we have qσ axσ(a) = NrdA (a)qσ (x) and

pσ axσ(a) = NrdA (a)σ(a)’1 pσ (x)a’1 .

(2) There exists some dσ ∈ F — such that

(a) qσ (x)2 = dσ NrdA (x) for all x ∈ Skew(A, σ);

(b) p2 = dσ · IdSkew(A,σ) ;

σ

(c) dσ · F —2 = disc σ;

(d) pσ (x), pσ (y) = dσ [x, y] for all x, y ∈ Skew(A, σ) where [ , ] are the Lie

brackets (i.e., [x, y] = xy ’ yx).

Proof : Consider θ, pθ , qθ as in (??). The relation

qθ (x)2 = NrdA (x) for x ∈ Skew(A, σ)

is easily proved: extending scalars to an algebraic closure, it su¬ces to show that

for any 2 — 2 matrices m1 , m2 of trace zero,

2

det(m1 — 1 + 1 — m2 ) = det(m1 ) ’ det(m2 ) .

This follows by a computation which is left to the reader. It is clear that p2 = θ

IdSkew(A,θ) , and (??) shows that disc θ = 1. For x1 , y1 ∈ (a1 , b1 )0 and x2 , y2 ∈

F

§16. BIQUATERNION ALGEBRAS 249

(a2 , b2 )0 , we have

F

[x1 — 1 + 1 — x2 , y1 — 1 + 1 — y2 ] = [x1 — 1, y1 — 1] + [1 — x2 , 1 — y2 ]

= [x1 — 1 ’ 1 — x2 , y1 — 1 ’ 1 — y2 ],

hence pθ (x), pθ (y) = [x, y] for x, y ∈ Skew(A, σ). Therefore, the properties in (??)

hold with dθ = 1. The relations

and pθ axθ(a) = NrdA (a)θ(a)’1 pθ (x)a’1

qθ axθ(a) = NrdA (a)qθ (x)

for all a ∈ A— and x ∈ Skew(A, σ) follow by the same arguments as in (??) and

(??). The proposition is thus proved for σ = θ.

For an arbitrary orthogonal involution σ, there exists v ∈ A— such that σ =

Int(v) —¦ θ. The proof of (??) then yields » ∈ F — such that

pσ (x) = »vpθ (xv) and qσ (x) = »qθ (xv)

for all x ∈ Skew(A, σ). For a ∈ A— and x ∈ Skew(A, σ), we then have

qσ axσ(a) = »qθ axvθ(a) and pσ axσ(a) = »vpθ axvθ(a) .

Since property (??) is already proved for θ, it follows that

qσ axσ(a) = » NrdA (a)qθ (xv) = NrdA (a)qσ (x)

and

pσ axσ(a) = » NrdA (a)vθ(a)’1 pθ (xv)a’1

= » NrdA (a)σ(a)’1 vpθ (xv)a’1

= NrdA (a)σ(a)’1 pσ (x)a’1 ,

proving(??).

To complete the proof, we show that the properties in (??) hold with dσ =

2

» NrdA (v).

First, we have for x ∈ Skew(A, σ) that

qσ (x)2 = »2 qθ (xv)2 = »2 NrdA (v) NrdA (x)

and

p2 (x) = »vpθ »vpθ (xv)v .

σ

Since θ(v) = v, we may use property(??) for θ to rewrite the right side as

»2 v NrdA (v)v ’1 p2 (xv)v ’1 = »2 NrdA (v)x.

θ

Therefore, (??) and (??) hold. Since

disc σ = NrdA (v) · disc θ = NrdA (v) · F —2 ,

by (??), we also have (??). Finally, to establish (??), observe that by linearizing

the relations

qσ (x) = xpσ (x) = pσ (x)x

we get

bqσ (x, y) = xpσ (y) + ypσ (x) = pσ (x)y + pσ (y)x for x, y ∈ Skew(A, σ).

In particular,

bqσ pσ (x), y = pσ (x)pσ (y) + yp2 (x) = p2 (x)y + pσ (y)pσ (x)

σ σ

250 IV. ALGEBRAS OF DEGREE FOUR

for x, y ∈ Skew(A, σ). In view of (??), it follows that

pσ (x)pσ (y) + dσ yx = dσ xy + pσ (y)pσ (x),

hence

pσ (x)pσ (y) ’ pσ (y)pσ (x) = dσ (xy ’ yx)

for x, y ∈ Skew(A, σ), proving (??).

Alternately, properties (??) and (??) can be established by comparing σ with

a symplectic involution instead of θ (see the proof of (??)), and using (??). Details

are left to the reader.

With the notation of the preceding proposition, we have for all x ∈ Skew(A, σ)

qσ pσ (x) = p2 (x)pσ (x) = dσ xpσ (x) = dσ qσ (x),

σ

hence pσ is a similitude of Skew(A, σ), qσ with multiplier dσ . The group of simil-

itudes of Skew(A, σ), qσ can be described by mimicking (??).

(16.26) Proposition. The proper similitudes of Skew(A, σ), qσ are of the form

x ’ »’1 axσ(a)

where » ∈ F — and a ∈ A— .

The improper similitudes of Skew(A, σ), qσ are of the form

x ’ »’1 apσ (x)σ(a)

where » ∈ F — and a ∈ A— .

The proof is left to the reader.

We now give another point of view on the linear endomorphism pσ and the

Albert form qσ by relating them to the Cli¬ord algebra C(A, σ).

√

Let K = F ( disc σ) and let ι be the nontrivial automorphism of K/F . Recall

from §?? that we may identify

(A, σ) = NK/F (Q, γ)

for some quaternion K-algebra Q with canonical involution γ. The quaternion

algebra Q is canonically isomorphic as an F -algebra to the Cli¬ord algebra C(A, σ).

Recall also that there is a Lie algebra homomorphism n : L(Q) ’ L(A) de¬ned by

™

n(x) = ι x — 1 + ι 1 — x for x ∈ Q.

™

This homomorphism restricts to a Lie algebra isomorphism

∼

n : Q0 ’ Skew(A, σ).

™ ’

(16.27) Proposition. Let ± ∈ K — be such that ι(±) = ’±. The linear endomor-

phism pσ which makes the following diagram commutative:

n

™

Q0 ’ ’ ’ Skew(A, σ)

’’

¦ ¦

¦ ¦ pσ

±·

n

™

Q0 ’ ’ ’ Skew(A, σ)

’’

(where ±· is multiplication by ±) is such that xpσ (x) ∈ F for all x ∈ Skew(A, σ).

The corresponding Albert form qσ satis¬es:

qσ n(x) = trK/F (±x2 ) for x ∈ Q0 .

™

§16. BIQUATERNION ALGEBRAS 251

Let s : Q0 ’ K be the squaring map de¬ned by s(x) = x2 . Then qσ is the Scharlau

transfer of the form ± · s with respect to the linear form trK/F : K ’ F :

qσ = (trK/F )— ± · s .

Proof : It su¬ces to prove that

n(x)n(±x) = trK/F (±x2 ) for x ∈ Q0 .

™ ™

This follows from a straightforward computation:

n(x)n(±x) = (ι x — 1 + 1 — x) ι (±x) — 1 + 1 — ±x

™ ™

= ι(±x2 ) + (±x2 ) + ι (±x) — x + ι x — ±x.

Since ι(±) = ’±, the last two terms in the last expression cancel.

Continuing with the notation of the proposition above and letting dσ = ±2 ∈

F — , we obviously have p2 = dσ IdSkew(A,σ) and dσ · F —2 = disc σ, and also

σ

pσ (x), pσ (y) = dσ [x, y] for x, y ∈ Skew(A, σ)

since n is an isomorphism of Lie algebras. We may thus recover the properties

™

in (??.??).

Conversely, Proposition (??) shows that the linear endomorphism pσ can be

used to endow Skew(A, σ) with a structure of K-module, hence to give an explicit

description of the Cli¬ord algebra C(A, σ).

We may also derive some information on quaternion algebras over quadratic

extensions:

(16.28) Corollary. For a quaternion algebra Q over an ´tale quadratic exten-

e

sion K/F , the following conditions are equivalent:

(1) Q is split by some quadratic extension of F ;

(2) Q (a, b)K for some a ∈ F — and some b ∈ K — ;

(3) ind NK/F (Q) = 1 or 2.

Proof : It su¬ces to prove the equivalence of (??) and (??) for non-split quaternion

algebras Q, since (??) and (??) are clearly equivalent and the three conditions

trivially hold if Q is split. We may thus assume that the squaring map s : Q0 ’ K

is anisotropic. Condition (??) then holds if and only if the transfer (tr K/F )— ± ·s

is isotropic where ± ∈ K is an arbitrary nonzero element of trace zero. By (??)

and (??), (trK/F )— ± · s is an Albert form of NK/F (Q); therefore, by Albert™s

Theorem (??), this form is isotropic if and only if condition (??) holds.

In the special case where K = F — F , the corollary takes the following form:

(16.29) Corollary (Albert [?]). For quaternion algebras Q1 , Q2 over F , the fol-

lowing conditions are equivalent:

(1) there is a quadratic extension of F which splits both Q1 and Q2 ;

(2) Q1 (a, b1 )F and Q2 (a, b2 )F for some a, b1 , b2 ∈ F — ;

(3) ind(Q1 —F Q2 ) = 1 or 2.

The implication (??) ’ (??) may be reformulated as follows:

(16.30) Corollary. Let Q1 , Q2 , Q3 be quaternion algebras over F . If Q1 —Q2 —Q3

is split, then there is a quadratic extension of F which splits Q1 , Q2 , and Q3 .

252 IV. ALGEBRAS OF DEGREE FOUR

Proof : The hypothesis means that Q1 — Q2 is Brauer-equivalent to Q3 , hence its

index is 1 or 2. The preceding corollary yields a quadratic extension of F which

splits Q1 and Q2 , hence also Q3 .

Finally, we relate the preceding constructions to the generalized pfa¬an de¬ned

in §??. Let Z(A, σ) be the center of the Cli¬ord algebra C(A, σ). Let ι be the

nontrivial automorphism of Z(A, σ)/F and let Z(A, σ)0 be the space of elements

of trace zero, so

Z(a, σ)0 = { z ∈ Z(A, σ) | ι(z) = ’z }.

By (??), the generalized pfa¬an is a quadratic map

π : Skew(A, σ) ’ Z(A, σ)0 .

Our goal is to show that this map can be regarded as a canonical Albert form

associated to σ.

Let C(A, σ)0 be the space of elements of reduced trace zero in C(A, σ). Since

C(A, σ) is a quaternion algebra over Z(A, σ), we have x2 ∈ Z(A, σ) for x ∈

C(A, σ)0 . We may then de¬ne a map • : C(A, σ)0 ’ Z(A, σ)0 by

•(x) = x2 ’ ι(x2 ) for x ∈ C(A, σ)0 .

(16.31) Lemma. Let c : A ’ C(A, σ) be the canonical map. For x ∈ Skew(A, σ),

1

π(x) = • 2 c(x) .

Proof : We may extend scalars to an algebraic closure, and assume that

(A, σ) = M4 (F ), t = EndF (F 4 ), σq

where q(x1 , x2 , x3 , x4 ) = x2 +x2 +x2 +x2 . We use q to identify EndF (F 4 ) = F 4 —F 4

1 2 3 4

as in (??). Let (ei )1¤i¤4 be the standard basis of F 4 . A basis of Skew(A, σ) is given

by

hij = 1 (ei — ej ’ ej — ei ) for 1 ¤ i < j ¤ 4,

2

and we have, by (??),

π( xij hij ) = (x12 x34 ’ x13 x24 + x14 x23 )e1 · e2 · e3 · e4

1¤i<j¤4

for xij ∈ F . On the other hand,

1 1

2 c( xij hij ) = xij ei · ej ,

1¤i<j¤4 1¤i<j¤4

2

and a computation shows that

•( xij hij ) = (x12 x34 ’ x13 x24 + x14 x23 )e1 · e2 · e3 · e4 .

1¤i<j¤4

Theorem (??) yields a canonical isomorphism

(A, σ) = NZ(A,σ)/F C(A, σ), σ ;

moreover, using the canonical isomorphism as an identi¬cation, the map

n : C(A, σ)0 ’ Skew NZ(A,σ)/F C(A, σ), σ

™

1

Skew(A, σ) ’ C(A, σ)0 (see (??)). Therefore, the lemma

is the inverse of 2c:

yields:

π n(x) = •(x) = x2 ’ ι(x2 ) for x ∈ C(A, σ)0 .

™

§17. WHITEHEAD GROUPS 253

Let ± ∈ Z(A, σ)0 , ± = 0. According to (??), the map qσ : Skew(A, σ) ’ F de¬ned

by

for x ∈ C(A, σ)0

qσ n(x) = n(x)n(±x)

™ ™ ™

is an Albert form of A.

(16.32) Proposition. For all a ∈ Skew(A, σ),

qσ (a) = ±π(a).

Proof : It su¬ces to prove that

n(x)n(±x) = ± x2 ’ ι(x2 ) for x ∈ C(A, σ)0 .

™ ™

The left side has been computed in the proof of (??):

n(x)n(±x) = ι(±x2 ) + ±x2 .

™ ™

§17. Whitehead Groups

For an arbitrary central simple algebra A, we set

SL1 (A) = { a ∈ A— | NrdA (a) = 1 }.

This group contains the normal subgroup [A— , A— ] generated by commutators

aba’1 b’1 . The factor group is denoted

SK1 (A) = SL1 (A)/[A— , A— ].

This group is known in algebraic K-theory as the reduced Whitehead group of A.

It is known that SK1 (A) = 0 if A is split (and A = M2 (F2 )) or if the index

of A is square-free (a result due to Wang [?], see for example Pierce [?, 16.6] or

the lecture notes [?]). In the ¬rst subsection, we consider the next interesting case

where A is a biquaternion algebra. Let F be the center of A, which may be of

arbitrary characteristic. Denote by I k F the k-th power of the fundamental ideal

IF of the Witt ring W F of nonsingular bilinear spaces over F , and let Wq F denote

the Witt group of nonsingular even-dimensional quadratic spaces over F , which

is a module over W F . We write I k Wq F for I k F · Wq F , so I k Wq F = I k+1 F if

char F = 2. Our objective is to de¬ne a canonical injective homomorphism

± : SK1 (A) ’ I 3 Wq F/I 4 Wq F,

from which examples where SK1 (A) = 0 are easily derived.

In the second subsection, we brie¬‚y discuss analogues of the reduced Whitehead

group for algebras with involution in characteristic di¬erent from 2.

17.A. SK1 of biquaternion algebras. Although the map ± that we will

de¬ne is canonical, it is induced by a map ±σ whose de¬nition depends on the choice

of a symplectic involution. Therefore, we start with some general observations on

symplectic involutions on biquaternion algebras.

Let A be a biquaternion algebra over a ¬eld F of arbitrary characteristic, and

let σ be a symplectic involution on A. Recall from §?? the linear endomorphism

of Symd(A, σ) de¬ned by x = Trpσ (x) ’ x, and the quadratic form

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

254 IV. ALGEBRAS OF DEGREE FOUR

As in §??, we let

Symd(A, σ)0 = { x ∈ Symd(A, σ) | Trpσ (x) = 0 },

and we write Symd(A, σ)— = Symd(A, σ) © A— for simplicity. For v ∈ Symd(A, σ)—

and x ∈ A, we have σ(x)vx ∈ Symd(A, σ), because if v = w + σ(w), then σ(x)vx =

σ(x)wx + σ σ(x)wx . We may therefore consider the quadratic form ¦v : A ’ F

de¬ned by

¦v (x) = Trpσ σ(x)vx for x ∈ A.

(17.1) Proposition. For each v ∈ Symd(A, σ)— , the quadratic form ¦v is a

scalar multiple of a 4-fold P¬ster form. This form is hyperbolic if Trpσ (v) = 0.

Moreover, if σ is a hyperbolic symplectic involution, then ¦v is hyperbolic for all

v ∈ Symd(A, σ)— .

Proof : Suppose ¬rst that σ is hyperbolic. The algebra A then contains an isotropic

right ideal I of reduced dimension 2, i.e., dimF I = 8. For x ∈ I we have σ(x)x = 0,

hence for all v = w + σ(w) ∈ Symd(A, σ)— ,

¦v σ(x) = Trpσ xvσ(x) = TrdA xwσ(x) = TrdA wσ(x)x = 0.

Therefore, σ(I) is a totally isotropic subspace of A for the form ¦v , hence this form

is hyperbolic.

For the rest of the proof, assume σ is not hyperbolic, and let V ‚ Symd(A, σ) be

a 3-dimensional subspace containing 1 and v, and not contained in Symd(A, σ)0 . By

(??), there is a decomposition of A into a tensor product of quaternion subalgebras,

so that

(A, σ) = (Q1 , σ1 ) —F (Q2 , γ2 )

where σ1 is an orthogonal involution, γ2 is the canonical involution, and v ∈

Sym(Q1 , σ1 ). In view of the computation of the bilinear form T(Q1 ,σ1 ,v) in (??), the

following lemma completes the proof:

(17.2) Lemma. Suppose (A, σ) = (Q1 , σ1 ) —F (Q2 , γ2 ), where σ1 is an orthogonal

involution and γ2 is the canonical (symplectic) involution. We have Sym(Q1 , σ1 ) ‚

Symd(A, σ) and, for all v ∈ Sym(Q1 , σ1 )— ,

¦v = T(Q1 ,σ1 ,v) · NrdQ2

where NrdQ2 is the reduced norm quadratic form on Q2 .

Proof : Let ∈ Q2 be such that + γ2 ( 2 ) = 1. For all s ∈ Sym(Q1 , σ1 ), we have

2 2

s—1=s— + σ(s — 2 ),

2

hence s — 1 ∈ Symd(A, σ) and

Trpσ (s — 1) = TrdA (s — 2) = TrdQ1 (s) TrdQ2 ( 2 ) = TrdQ1 (s).

Let v ∈ Sym(Q1 , σ1 )— . For x1 ∈ Q1 and x2 ∈ Q2 , we have

σ(x1 — x2 )v(x1 — x2 ) = σ1 (x1 )vx1 — γ2 (x2 )x2 = σ1 (x1 )vx1 — 1 NrdQ2 (x2 ).

Therefore,

¦v (x1 — x2 ) = Trpσ σ1 (x1 )vx1 — 1 NrdQ2 (x2 ) = TrdQ1 σ1 (x1 )vx1 NrdQ2 (x2 ),

hence

¦v (x1 — x2 ) = T(Q1 ,σ1 ,v) (x1 , x1 ) NrdQ2 (x2 ).

§17. WHITEHEAD GROUPS 255

To complete the proof, it remains only to show that the polar form b¦v of ¦v is the

tensor product T(Q1 ,σ1 ,v) — bNrdQ2 .

For x, y ∈ A, we have

b¦v (x, y) = Trpσ σ(x)vy + σ σ(x)vy = TrdA σ(x)vy ,

hence, for x = x1 — x2 and y = y1 — y2 ,

b¦v (x, y) = TrdQ1 σ1 (x1 )vy1 TrdQ2 γ2 (x2 )y2 .

Since TrdQ2 γ2 (x2 )y2 = bNrdQ2 (x2 , y2 ), the proof is complete.

The de¬nition of ±σ uses the following result, which is reminiscent of Hilbert™s

theorem 90:

(17.3) Lemma. Suppose σ is not hyperbolic. For every u ∈ Symd(A, σ) such that

Nrpσ (u) = 1, there exists v ∈ Symd(A, σ)— such that u = vv ’1 . If u = ’1, the