— 0 a

hence f is proper. Similarly, the same arguments show that the similitudes x ’

»’1 axσ(a) are improper.

Returning to the case where f is an arbitrary similitude of Symd(A, σ), Nrpσ

and f— —¦ i’1 = Int(u) with u = v w or w v , we apply f— —¦ i’1 to i(x) for

0 0

— —

0 0

x ∈ Symd(A, σ) and get

±’1 f (x) 0 x ’1

0

(16.12) =u u.

x0

f (x) 0

Comparing the lower left corners yields that

wxv ’1 if f is proper,

(16.13) f (x) =

wxv ’1 if f is improper.

Let θ be the involution on M2 (A) de¬ned by

a11 a12 σ(a22 ) ’σ(a12 )

θ = .

a21 a22 ’σ(a21 ) σ(a11 )

Applying θ to both sides of (??), we get

±’1 f (x) 0x

0

= θ(u)’1 θ(u).

x0

f (x) 0

Therefore, θ(u)u commutes with the matrices of the form i(x) for x ∈ Symd(A, σ).

Since these matrices generate M2 (A), it follows that θ(u)u ∈ F — , hence

σ(w)v = σ(v)w ∈ F — .

Letting σ(w)v = », we derive from (??) that

»’1 wxσ(w) if f is proper,

f (x) =

»’1 wxσ(w) if f is improper.

Since the multipliers of the similitudes x ’ »’1 axσ(a) and x ’ »’1 axσ(a)

are »’2 NrdA (a), the multipliers of the Albert form Nrpσ are

G(Nrpσ ) = F —2 · NrdA (A— ).

We thus get another proof of the ¬rst part of (??).

(16.14) Example. Suppose Q is a quaternion algebra over F , with canonical in-

volution γ, and A = M2 (Q) with the involution σ de¬ned by σ (qij )1¤i,j¤2 =

t

γ(qij ) 1¤i,j¤2 ; then

±11 a12

Symd(A, σ) = ±11 , ±22 ∈ F , a12 ∈ Q .

γ(a12 ) ±22

±11 a12

For a = ∈ Symd(A, σ), we have Trpσ (a) = ±11 + ±22 , hence a =

γ(a12 ) ±22

±22 ’a12

and therefore

’γ(a12 ) ±11

Nrpσ (a) = ±11 ±22 ’ NrdQ (a12 ).

240 IV. ALGEBRAS OF DEGREE FOUR

This expression is the Moore determinant of the hermitian matrix a (see Jacobson

[?]). This formula shows that the matrices in Symd(A, σ) whose diagonal entries

vanish form a quadratic space isometric to (Q, ’ NrdQ ). On the other hand, the

diagonal matrices form a hyperbolic plane H, and

H ⊥ ’ NrdQ .

Nrpσ

Also, for the involution θ de¬ned by

a11 a12 γ(a11 ) ’γ(a21 )

θ = ,

a21 a22 ’γ(a12 ) γ(a22 )

we have

±11 a12

Symd(A, θ) = ±11 , ±22 ∈ F , a12 ∈ Q .

’γ(a12 ) ±22

±11 a12

For a = ∈ Symd(A, σ), we get

’γ(a12 ) ±22

Nrpθ (a) = ±11 ±22 + NrdQ (a12 ),

hence

H ⊥ NrdQ .

Nrpθ

A more general example is given next.

(16.15) Example. Suppose A = Q1 — Q2 is a tensor product of quaternion al-

gebras Q1 , Q2 with canonical (symplectic) involutions γ1 , γ2 . Let v1 be a unit

in Skew(Q1 , γ1 ) and σ1 = Int(v1 ) —¦ γ1 . The involution σ1 on Q1 is orthogonal,

unless v1 ∈ F — , a case which occurs only if char F = 2. Therefore, the involution

σ = σ1 — γ2 on A is symplectic in all cases, by (??). Our goal is to compute

explicitly the quadratic form Nrpσ .

As a ¬rst step, observe that

Alt(A, γ1 — γ2 ) = { x1 — 1 ’ 1 — x2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) },

as pointed out in Exercise ?? of Chapter ??; therefore,

Symd(A, σ) = (v1 — 1) · Alt(A, γ1 — γ2 )

= { v1 x1 — 1 ’ v1 — x2 | TrdQ1 (x1 ) = TrdQ2 (x2 ) }.

For x1 ∈ Q1 and x2 ∈ Q2 such that TrdQ1 (x1 ) = TrdQ2 (x2 ), there exist y1 ∈ Q1 ,

y2 ∈ Q2 such that x1 = TrdQ2 (y2 )y1 and x2 = TrdQ1 (y1 )y2 (see (??)), hence

x1 — 1 ’ 1 — x2 = y1 — γ2 (y2 ) ’ γ1 (y1 ) — y2

and therefore

v1 x1 — 1 ’ v1 — x2 = v1 y1 — γ2 (y2 ) + σ v1 y1 — γ2 (y2 ) .

By (??), it follows that

Trpσ (v1 x1 — 1 ’ v1 — x2 ) = TrdA v1 y1 — γ2 (y2 ) = TrdQ1 (v1 y1 ) TrdQ2 (y2 ).

Since TrdQ2 (y2 )y1 = x1 , we get

Trpσ (v1 x1 — 1 ’ v1 — x2 ) = TrdQ1 (v1 x1 ),

hence

v1 x1 — 1 ’ v1 — x2 = γ1 (v1 x1 ) — 1 + v1 — x2

§16. BIQUATERNION ALGEBRAS 241

and ¬nally

Nrpσ (v1 x1 — 1 ’ v1 — x2 ) = NrdQ1 (v1 ) NrdQ1 (x1 ) ’ NrdQ2 (x2 ) .

This shows that the form Nrpσ on Symd(A, σ) is similar to the quadratic form

qγ1 —γ2 on Alt(A, γ1 — γ2 ) de¬ned by

qγ1 —γ2 (x1 — 1 ’ 1 — x2 ) = NrdQ1 (x1 ) ’ NrdQ2 (x2 )

for x1 ∈ Q1 , x2 ∈ Q2 such that TrdQ1 (x1 ) = TrdQ2 (x2 ).

To give a more explicit description of qγ1 —γ2 , we consider the case where

char F = 2 separately. Suppose ¬rst char F = 2, and let Q1 = (a1 , b1 )F , Q2 =

(a2 , b2 )F with quaternion bases (1, i1 , j1 , k1 ) and (1, i2 , j2 , k2 ) respectively. Then

(i1 — 1, j1 — 1, k1 — 1, 1 — i2 , 1 — j2 , 1 — k2 ) is an orthogonal basis of Alt(A, γ1 — γ2 )

which yields the following diagonalization of qγ1 —γ2 :

qγ1 —γ2 = ’a1 , ’b1 , a1 b1 , a2 , b2 , ’a2 b2

(compare with (??) and (??)); therefore,

Nrpσ NrdQ1 (v1 ) · ’a1 , ’b1 , a1 b1 , a2 , b2 , ’a2 b2 .

Suppose next that char F = 2, and let Q1 = [a1 , b1 )F , Q2 = [a2 , b2 )F with quater-

nion bases (in characteristic 2) (1, u1 , v1 , w1 ) and (1, u2 , v2 , w2 ) respectively. A basis

of Alt(A, γ1 — γ2 ) is (1, u1 — 1 + 1 — u2 , v1 — 1, w1 — 1, 1 — v2 , 1 — w2 ). With respect

to this basis, the form qγ1 —γ2 has the following expression:

qγ1 —γ2 = [1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ]

(compare with (??)); therefore,

Nrpσ NrdQ1 (v1 ) · [1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ] .

The following proposition yields a decomposition of the type considered in the

example above for any biquaternion algebra with symplectic involution; it is thus

an explicit version of the second proof of (??). However, for simplicity we restrict to

symplectic involutions which are not hyperbolic.25 In view of (??), this hypothesis

means that the space

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

does not contain any nonzero vector x such that x2 = 0 or, equivalently, Nrpσ (x) =

0. Therefore, all the nonzero elements in Symd(A, σ)0 are invertible.

(16.16) Proposition. Let (A, σ) be a biquaternion algebra with symplectic invo-

lution over an arbitrary ¬eld F . Assume that σ is not hyperbolic, and let V ‚

Symd(A, σ) be a 3-dimensional subspace such that

F ‚ V ‚ Symd(A, σ)0 .

Then there exists a unique quaternion subalgebra Q1 ‚ A containing V . This

quaternion algebra is stable under σ, and the restriction σ1 = σ|Q1 is orthogonal.

Therefore, for Q2 = CA Q1 the centralizer of Q1 , we have

(A, σ) = (Q1 , σ1 ) — (Q2 , γ2 )

where γ2 is the canonical involution on Q2 .

25 See Exercise ?? of Chapter ?? for the hyperbolic case.

242 IV. ALGEBRAS OF DEGREE FOUR

Proof : Choose v ∈ V F such that Trpσ (v) = 0; then F [v] = F • vF is an ´talee

quadratic F -subalgebra of A, and restricts to the nontrivial automorphism of

F [v]. Pick a nonzero vector u ∈ V © Symd(A, σ)0 which is orthogonal to v for

the polar form bNrpσ ; we then have uv + vu = 0, which means that uv = vu,

since Trpσ (u) = 0. The hypothesis that σ is not hyperbolic ensures that u is

invertible; therefore, u and v generate a quaternion subalgebra Q1 = F [v], u2 F .

This quaternion subalgebra contains V , and is indeed generated by V . Since u and

v are symmetric under σ, it is stable under σ, and Sym(Q1 , σ1 ) = V . Moreover,

Sym(Q1 , σ1 ) contains the element v such that TrdQ1 (v) = v + v = Trpσ (v) = 0,

hence σ1 is orthogonal. The rest follows from (??) and (??).

The invariant of symplectic involutions. Let σ be a ¬xed symplectic in-

volution on a biquaternion algebra A. To every other symplectic involution „ on A,

we associate a quadratic form jσ („ ) over F which classi¬es symplectic involutions

up to conjugation: jσ („ ) jσ („ ) if and only if „ = Int(a) —¦ „ —¦ Int(a)’1 for some

a ∈ A— .

We ¬rst compare the Albert forms Nrpσ and Nrp„ associated with symplectic

involutions σ and „ . Recall from (??) that „ = Int(u) —¦ σ for some unit u ∈

Symd(A, σ). Multiplication on the left by the element u then de¬nes a linear map

∼

Symd(A, σ) ’ Symd(A, „ ).

’

(16.17) Lemma. For all x ∈ Symd(A, σ),

Nrp„ (ux) = Nrpσ (u) Nrpσ (x).

Proof : Both sides of the equation to be established are quadratic forms on the space

Symd(A, σ). These quadratic forms di¬er at most by a factor ’1, since squaring

both sides yields the equality

NrdA (ux) = NrdA (u) NrdA (x).

On the other hand, for x = 1 these quadratic forms take the same nonzero value

since from the fact that PrdA,u = Prp2 = Prp2 it follows that Prpσ,u = Prp„,u ,

σ,u „,u

hence Nrpσ (u) = Nrp„ (u). Therefore, the quadratic forms are equal.

Let W F denote the Witt ring of nonsingular bilinear forms over F and write

Wq F for the W F -module of even-dimensional nonsingular quadratic forms. For

every integer k, the k-th power of the fundamental ideal IF of even-dimensional

forms in W F is denoted I k F ; we write I k Wq F for the product I k F · Wq F . Thus, if

I k Wq F = I k+1 F if char F = 2. From the explicit formulas in (??), it is clear that

Albert forms are in IWq F ; indeed, if char F = 2,

a1 , b1 , ’a1 b1 , ’a2 , ’b2 , a2 b2 = ’ a1 , b1 + a2 , b2 in W F,

and, if char F = 2,

[1, a1 + a2 ] ⊥ b1 · [1, a1 ] ⊥ b2 · [1, a2 ] = b1 , a1 ]] + b2 , a2 ]] in Wq F.

(16.18) Proposition. Let σ, „ be symplectic involutions on a biquaternion F -

algebra A and let „ = Int(u) —¦ σ for some unit u ∈ Symd(A, σ). In the Witt

group Wq F ,

Nrpσ (u) · Nrpσ = Nrpσ ’ Nrp„ .

There is a 3-fold P¬ster form jσ („ ) ∈ I 2 Wq F and a scalar » ∈ F — such that

» · jσ („ ) = Nrpσ (u) · Nrpσ in Wq F .

§16. BIQUATERNION ALGEBRAS 243

The P¬ster form jσ („ ) is uniquely determined by the condition

mod I 3 Wq F.

jσ („ ) ≡ Nrpσ (u) · Nrpσ

Proof : Lemma (??) shows that multiplication on the left by u is a similitude:

Symd(A, σ), Nrpσ ’ Symd(A, „ ), Nrp„

with multiplier Nrpσ (u). Therefore, Nrp„ Nrpσ (u) · Nrpσ , hence

Nrpσ ’ Nrp„ Nrpσ (u) · Nrpσ .

We next show the existence of the 3-fold P¬ster form jσ („ ). Since 1 and

u are anisotropic for Nrpσ , there exist nonsingular 3-dimensional subspaces U ‚

Symd(A, σ) containing 1 and u. Choose such a subspace and let qU be the restriction

of Nrpσ to U . Let q0 be a 4-dimensional form in IWq F containing qU as a subspace:

if char F = 2 and qU a1 , a2 , a3 , we set q0 = a1 , a2 , a3 , a1 a2 a3 ; if char F = 2

[a1 , a2 ] ⊥ [a3 ], we set q0 = [a1 , a2 ] ⊥ [a3 , a1 a2 a’1 ]. Since the quadratic

and qU 3

forms Nrpσ and q0 have isometric 3-dimensional subspaces, there is a 4-dimensional

quadratic form q1 such that

q0 + q1 = Nrpσ in Wq F.

The form q1 lies in IWq F , since q0 and Nrpσ are in this subgroup. Moreover, since

Nrpσ (u) is represented by qU , hence also by q0 , we have Nrpσ (u) ·q0 = 0 in Wq F .

Therefore, multiplying both sides of the equality above by Nrp σ (u) , we get

Nrpσ (u) · q1 = Nrpσ (u) · Nrpσ in Wq F.

The form on the left is a scalar multiple of a 3-fold P¬ster form which may be

chosen for jσ („ ). This P¬ster form satis¬es jσ („ ) ≡ Nrpσ (u) · q1 mod I 3 Wq F ,

hence also

mod I 3 Wq F.

jσ („ ) ≡ Nrpσ (u) · Nrpσ

It remains only to show that it is uniquely determined by this condition. This

follows from the following general observation: if π, π are k-fold P¬ster forms such

that

mod I k Wq F,

π≡π

then the di¬erence π ’ π is represented by a quadratic form of dimension 2k+1 ’ 2

since π and π both represent 1. On the other hand, π ’ π ∈ I k Wq F , hence the

Hauptsatz of Arason and P¬ster (see26 Lam [?, p. 289] or Scharlau [?, Ch. 4, §5])

shows that π ’ π = 0. Therefore, π π .

We next show that the invariant jσ classi¬es symplectic involutions up to con-

jugation:

(16.19) Theorem. Let σ, „ , „ be symplectic involutions on a biquaternion algebra

A over an arbitrary ¬eld F , and let „ = Int(u) —¦ σ, „ = Int(u ) —¦ σ for some units

u, u ∈ Symd(A, σ). The following conditions are equivalent:

(1) „ and „ are conjugate, i.e., there exists a ∈ A— such that „ = Int(a) —¦ „ —¦

Int(a)’1 ;

(2) Nrpσ (u) Nrpσ (u ) ∈ F —2 · Nrd(A— );

26 The proofs given there are easily adapted to the characteristic 2 case. The main ingredient

is the Cassels-P¬ster subform theorem, of which a characteristic 2 analogue is given in P¬ster [?,

Theorem 4.9, Chap. 1].

244 IV. ALGEBRAS OF DEGREE FOUR

(3) Nrpσ (u) Nrpσ (u ) ∈ G(Nrpσ );

(4) jσ („ ) jσ („ );

(5) Nrp„ Nrp„ .

Proof : (??) ’ (??) If there exists some a ∈ A— such that „ = Int(a)—¦„ —¦Int(a)’1 ,

then, since the right-hand side is also equal to Int a„ (a) —¦ „ , we get

Int(u ) = Int a„ (a)u = Int auσ(a) ,

hence u = »’1 auσ(a) for some » ∈ F — . By (??), it follows that

Nrpσ (u ) = »’2 NrdA (a) Nrpσ (u),

proving (??).

(??) ⇐’ (??) This readily follows from (??), since Nrpσ is an Albert form

of A.

(??) ’ (??) Suppose Nrpσ (u ) = µ Nrpσ (u) for some µ ∈ G(Nrpσ ). We may

then ¬nd a proper similitude g ∈ GO+ (Nrpσ ) with multiplier µ. Then

Nrpσ g(u) = µ Nrpσ (u) = Nrpσ (u ),

hence there is a proper isometry h ∈ O+ (Nrpσ ) such that h —¦ g(u) = u . By (??),

we may ¬nd an a ∈ A— and a » ∈ F — such that

h —¦ g(x) = »’1 axσ(a) for all x ∈ Symd(A, σ).

In particular, u = »’1 auσ(a), hence Int(a) —¦ „ = „ —¦ Int(a).

(??) ⇐’ (??) Since jσ („ ) and jσ („ ) are the unique 3-fold P¬ster forms which

are equivalent modulo I 3 Wq F to Nrpσ (u) · Nrpσ and Nrpσ (u ) · Nrpσ respec-

tively, we have jσ („ ) jσ („ ) if and only if Nrpσ (u) · Nrpσ ≡ Nrpσ (u ) · Nrpσ

mod I 3 Wq F . Using the relation Nrpσ (u) ’ Nrpσ (u ) ≡ Nrpσ (u) Nrpσ (u )

mod I 2 F , we may rephrase the latter condition as

Nrpσ (u) Nrpσ (u ) · Nrpσ ∈ I 3 Wq F.

By the Arason-P¬ster Hauptsatz, this relation holds if and only if

Nrpσ (u) Nrpσ (u ) · Nrpσ = 0,

which means that Nrpσ (u) Nrpσ (u ) ∈ G(Nrpσ ).

(??) ⇐’ (??) The relations

mod I 3 Wq F mod I 3 Wq F

jσ („ ) ≡ Nrpσ ’ Nrp„ and jσ („ ) ≡ Nrpσ ’ Nrp„

jσ („ ) if and only if Nrp„ ’ Nrp„ ∈ I 3 Wq F . By the Arason-

show that jσ („ )

P¬ster Hauptsatz, this relation holds if and only if Nrp„ ’ Nrp„ = 0.

(16.20) Remark. Theorem (??) shows that the conditions in (??) are also equiv-

alent to: Symd(A, „ )0 , s„ Symd(A, „ )0 , s„ .

(16.21) Example. As in (??), consider a quaternion F -algebra Q with canonical

involution γ, and A = M2 (Q) with the involution σ de¬ned by σ (qij )1¤i,j¤2 =

t

γ(qij ) . Let „ = Int(u) —¦ σ for some invertible matrix u ∈ Symd(A, σ). As

observed in (??), we have Nrpσ H ⊥ ’ NrdQ ; therefore,

jσ („ ) = Nrpσ (u) · NrdQ .

§16. BIQUATERNION ALGEBRAS 245

Since jσ („ ) is an invariant of „ , the image of Nrpσ (u) in F — / NrdQ (Q— ) also is an

invariant of „ up to conjugation. In fact, since Nrpσ (u) is the Moore determinant

of u, this image is the Jacobson determinant of the hermitian form

y1

h (x1 , x2 ), (y1 , y2 ) = γ(x1 ) γ(x2 ) · u ·

y2

on the 2-dimensional Q-vector space Q2 . (See the notes of Chapter ??.)

Of course, if Q is split, then NrdQ is hyperbolic, hence jσ („ ) = 0 for all sym-

plectic involutions σ, „ . Therefore, all the symplectic involutions are conjugate in

this case. (This is clear a priori, since all the symplectic involutions on a split

algebra are hyperbolic.)

16.C. Albert forms and orthogonal involutions. Let σ be an orthogonal

involution on a biquaternion F -algebra A. Mimicking the construction of the Albert

form associated to a symplectic involution, in this subsection we de¬ne a quadratic

form qσ on the space Skew(A, σ) in such a way that Skew(A, σ), qσ is an Albert

quadratic space. By contrast with the symplectic case, the form qσ is only de¬ned

up to a scalar factor, however, and our discussion is restricted to the case where the

characteristic is di¬erent from 2. We also show how the form qσ is related to the

norm form of the Cli¬ord algebra C(A, σ) and to the generalized pfa¬an de¬ned

in §??.

Throughout this subsection, we assume that char F = 2.

(16.22) Proposition. There exists a linear endomorphism

pσ : Skew(A, σ) ’ Skew(A, σ)

which satis¬es the following two conditions:

(1) xpσ (x) = pσ (x)x ∈ F for all x ∈ Skew(A, σ);

(2) an element x ∈ Skew(A, σ) is invertible if and only if xpσ (x) = 0.

The endomorphism pσ is uniquely determined up to a factor in F — . More precisely,

if pσ : Skew(A, σ) ’ Skew(A, σ) is a linear map such that xpσ (x) ∈ F for all

x ∈ Skew(A, σ) (or pσ (x)x ∈ F for all x ∈ Skew(A, σ)), then

pσ = »pσ

for some » ∈ F .

Proof : By (??), the intersection Skew(A, σ) © A— is nonempty. Let u be a skew-

symmetric unit and „ = Int(u) —¦ σ. The involution „ is symplectic by (??), and we

have

Sym(A, „ ) = u · Skew(A, σ) = Skew(A, σ) · u’1 .

Therefore, for x ∈ Skew(A, σ) we may consider ux ∈ Sym(A, „ ) where

: Sym(A, „ ) ’ Sym(A, „ )

is as in (??), and set

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

We have pσ (x)x = uxux = Nrp„ (ux) ∈ F and

xpσ (x) = u’1 (uxux)u = u’1 Nrp„ (ux)u = Nrp„ (ux),

hence pσ satis¬es (??). It also satis¬es (??), since Nrp„ (ux)2 = NrdA (ux).

246 IV. ALGEBRAS OF DEGREE FOUR

In order to make this subsection independent of §??, we give an alternate proof

of the existence of pσ . Consider an arbitrary decomposition of A into a tensor

product of quaternion subalgebras:

A = Q 1 —F Q2

and let θ = γ1 — γ2 be the tensor product of the canonical (conjugation) involutions

on Q1 and Q2 . The involution θ is orthogonal since char F = 2, and

Skew(A, θ) = (Q0 — 1) • (1 — Q0 ),

1 2

where Q0 and Q0 are the spaces of pure quaternions in Q1 and Q2 respectively.

1 2

De¬ne a map pθ : Skew(A, θ) ’ Skew(A, θ) by

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

for x1 ∈ Q0 and x2 ∈ Q0 . For x = x1 — 1 + 1 — x2 ∈ Skew(A, θ) we have

1 2

xpθ (x) = pθ (x)x = x2 ’ x2 = ’ NrdQ1 (x1 ) + NrdQ2 (x2 ) ∈ F,

1 2

hence (??) holds for pθ . If xpθ (x) = 0, then x is clearly invertible. Conversely,

if x is invertible and xpθ (x) = 0, then pθ (x) = 0, hence x = 0, a contradiction.

Therefore, pθ also satis¬es (??).

If σ is an arbitrary orthogonal involution on A, we have σ = Int(v) —¦ θ for some

v ∈ Sym(A, θ) © A— , by (??). We may then set

pσ (x) = vpθ (xv) for x ∈ Skew(A, σ)

and verify as above that pσ satis¬es the required conditions.

We next prove uniqueness of pσ up to a scalar factor. The following arguments

are based on Wadsworth [?]. For simplicity, we assume that F has more than three

elements; the result is easily checked when F = F3 .

Let pσ be a map satisfying (??) and (??), and let pσ : Skew(A, σ) ’ Skew(A, σ)

be such that xpσ (x) ∈ F for all x ∈ Skew(A, σ). For x ∈ Skew(A, σ), we let

qσ (x) = xpσ (x) ∈ F and qσ (x) = xpσ (x) ∈ F.

Let x ∈ Skew(A, σ) © A— ; we have pσ (x) = qσ (x)x’1 and pσ (x) = qσ (x)x’1 , hence

qσ (x)

pσ (x) = pσ (x).

qσ (x)

Suppose y is another unit in Skew(A, σ), and that it is not a scalar multiple of x.

We also have pσ (y) = qσ (y)y ’1 , hence pσ (y) is not a scalar multiple of pσ (x), and

q (y)

pσ (y) = qσ (y) pσ (y). We may ¬nd some ± ∈ F — such that x + ±y ∈ A— , since the

σ

equation in ±

qσ (x + ±y) = qσ (x) + ±qσ (x, y) + ±2 qσ (y) = 0

has at most two solutions and F has more than three elements. For this choice

of ±, we have

qσ (x + ±y)

pσ (x + ±y) = pσ (x + ±y).

qσ (x + ±y)

By linearity of pσ and pσ , it follows that

qσ (x + ±y) q (x + ±y)

pσ (x) + ± σ

pσ (x) + ±pσ (y) = pσ (y).

qσ (x + ±y) qσ (x + ±y)

§16. BIQUATERNION ALGEBRAS 247

On the other hand, we also have

qσ (x) q (y)

pσ (x) + ± σ

pσ (x) + ±pσ (y) = pσ (y),

qσ (x) qσ (y)

hence

qσ (x) q (x + ±y) q (y)

=σ =σ

qσ (x) qσ (x + ±y) qσ (y)

since pσ (x) and pσ (y) are linearly independent.

To conclude, observe that Skew(A, σ) has a basis (ei )1¤i¤6 consisting of in-

vertible elements: this is clear if σ is the involution θ de¬ned above, and it follows

for arbitrary σ since Skew(A, σ) = v · Skew(A, θ) if σ = Int(v) —¦ θ. Denoting

» = qσ (e1 )qσ (e1 )’1 , the argument above shows that

qσ (ei )