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t
a nonsingular alternating matrix C ∈ M2g (Q) such that P CP t = 0 and iP CP is
positive de¬nite hermitian. The study of correspondences on the Riemann surface
leads one to consider the matrices M ∈ M2g (Q) for which there exists a matrix
68 I. INVOLUTIONS AND HERMITIAN FORMS


K ∈ Mg (C) such that KP = P M . Following Weyl™s simpler formulation [?],
P
one considers the matrix W = ∈ M2g (C) and the so-called Riemann matrix
P
’iIg 0
R = W ’1 W ∈ M2g (R). The matrices M such that KP = P M for
0 iIg
some K ∈ Mg (C) are exactly those which commute with R. They form a subal-
gebra of M2g (Q) known as the multiplication algebra. As observed by Rosati [?],
this algebra admits the involution X ’ C ’1 X t C (see Weyl [?]). Albert completely
determined the structure of the multiplication algebra in three papers in the An-
nals of Mathematics in 1934“1935. An improved version, [?], see also [?], laid the
foundations of the theory of simple algebras with involutions.
Corollary (??) was observed independently by several authors: see Tits [?,
Proposition 3], Platonov [?, Proposition 5] and Rowen [?, Proposition 5.3].
The original proof of Albert™s theorem on quaternion algebras with involution
of the second kind (??) is given in [?, Theorem 10.21]. This result will be put in a
broader perspective in §??: the subalgebra Q0 is the discriminant algebra of (Q, σ)
(see (??)).
There is an extensive literature on Lie and Jordan structures in rings with
involution; we refer the reader to Herstein™s monographs [?] and [?]. In particu-
lar, Lemma (??) can be proved by ring-theoretic arguments which do not involve
scalar extension (and therefore hold for more general simple rings): see Herstein [?,
Theorem 2.2, p. 28]. In the same spirit, extension of Lie isomorphisms has been
investigated for more general rings: see11 Martindale [?], Rosen [?] and Beidar-
Martindale-Mikhalev [?].
§??. Part (??) of Theorem (??) is due to Albert [?, Theorem 10.19]. Albert
also proved part (??) for crossed products of a special kind: Albert assumes in [?,
Theorem 10.16] the existence of a splitting ¬eld of the form L —F K where L is
Galois over F . Part (??) was stated in full generality by Riehm [?] and proved by
Scharlau [?] (see also [?, §8.9]). In order to see that every central simple algebra
which is Brauer-equivalent to an algebra with involution also has an involution, it
is not essential to use (??): see Albert [?, Theorem 10.12] or Scharlau [?, Corol-
lary 8.8.3]. By combining this result with Exercises ?? and ??, we obtain an
alternate proof of Theorem (??).
§??. If E and E are Brauer-equivalent central simple F -algebras, then Morita
theory yields an E-E -bimodule P and an E -E-bimodule Q such that P —E Q E
and Q —E P E . If M is a right E-module, then there is a natural isomorphism
EndE (M ) = EndE (M —E P ).
Therefore, if E (hence also E ) has an involution, (??) yields one-to-one correspon-
dences between hermitian or skew-hermitian forms on M (up to a central factor),
involutions on EndE (M ) = EndE (M —E P ) and hermitian or skew-hermitian forms
on M —E P (up to a central factor). The correspondence between hermitian forms
can be made more precise and explicit; it is part of a Morita equivalence between
categories of hermitian modules which is discussed in Knus™ book [?, §1.9].
The notion of involution trace was introduced by Fr¨hlich and McEvett [?,
o
§7]. Special cases of the extension theorem (??) have been proved by Rowen [?,
Corollary 5.5] and by Lam-Leep-Tignol [?, Proposition 5.1]. Kneser™s theorem has

11 We are indebted to W. S. Martindale III for references to the recent literature.
NOTES 69


been generalized by Held and Scharlau [?] to the case where the subalgebra is
semisimple. (A particular case of this situation had also been considered by Kneser
in [?, p. 37].)
The existence of involutions for which a given element is symmetric or skew-
symmetric is discussed in Shapiro [?], which also contains an extensive discussion
of the literature.
§??. The de¬nition of quadratic pair in (??) is new. While involutions on
arbitrary central simple algebras have been related to hermitian forms in §??, the
relation between quadratic pairs and quadratic forms is described only in the split
case. The nonsplit case requires an extension of the notion of quadratic form.
For quaternion algebras such an extension was given by Seip-Hornix [?]. Tits [?]
de¬nes a (generalized) quadratic form as an element in the factor group A/ Alt(A, „ )
(compare with (??)); a more geometric viewpoint which also extends this notion
further, was proposed by Bak [?] (see also for instance Hahn-O™Meara [?, 5.1C],
Knus [?, Ch. 1, §5] or Scharlau [?, Ch. 7, §3]).
70 I. INVOLUTIONS AND HERMITIAN FORMS
CHAPTER II


Invariants of Involutions

In this chapter, we de¬ne various kinds of invariants of central simple algebras
with involution (or with quadratic pair) and we investigate their basic properties.
The invariants considered here are analogues of the classical invariants of quadratic
forms: the Witt index, the discriminant, the Cli¬ord algebra and the signature.
How far the analogy can be pushed depends of course on the nature of the in-
volution: an index is de¬ned for arbitrary central simple algebras with involution
or quadratic pair, but the discriminant is de¬ned only for orthogonal involutions
and quadratic pairs, and the Cli¬ord algebra just for quadratic pairs. The Cli¬ord
algebra construction actually splits into two parts: while it is impossible to de¬ne a
full Cli¬ord algebra for quadratic pairs, the even and the odd parts of the Cli¬ord
algebra can be recovered in the form of an algebra and a bimodule. For unitary
involutions, the notion of discriminant turns out to lead to a rich structure: we as-
sociate in §?? a discriminant algebra (with involution) to every unitary involution
on a central simple algebra of even degree. Finally, signatures can be de¬ned for
arbitrary involutions through the associated trace forms. These trace forms also
have relations with the discriminant or discriminant algebra. They yield higher
invariants for algebras with unitary involution of degree 3 in Chapter ?? and for
Jordan algebras in Chapter ??.
The invariants considered in this chapter are produced by various techniques.
The index is derived from a representation of the algebra with involution as the
endomorphism algebra of some hermitian or skew-hermitian space over a division
algebra, while the de¬nitions of discriminant and Cli¬ord algebra are based on
the fact that scalar extension reduces the algebra with quadratic pair to the endo-
morphism algebra of a quadratic space. We show that the discriminant and even
Cli¬ord algebra of the corresponding quadratic form can be de¬ned in terms of
the adjoint quadratic pair, and that the de¬nitions generalize to yield invariants
of arbitrary quadratic pairs. A similar procedure is used to de¬ne the discrimi-
nant algebra of a central simple algebra of even degree with unitary involution.
Throughout most of this chapter, our method of investigation is thus based on
scalar extension: after specifying the de¬nitions “rationally” (i.e., over an arbitrary
base ¬eld), the main properties are proven by extending scalars to a splitting ¬eld.
This method contrasts with Galois descent, where constructions over a separable
closure are shown to be invariant under the action of the absolute Galois group and
are therefore de¬ned over the base ¬eld.

§6. The Index
According to (??), every central simple F -algebra with involution (A, σ) can be
represented as EndD (V ), σh for some division algebra D, some D-vector space V
and some nonsingular hermitian form h on V . Since this representation is essentially
71
72 II. INVARIANTS OF INVOLUTIONS


unique, it is not di¬cult to check that the Witt index w(V, h) of the hermitian space
(V, h), de¬ned as the maximum of the dimensions of totally isotropic subspaces
of V , is an invariant of (A, σ). In this section, we give an alternate de¬nition of this
invariant which does not depend on a representation of (A, σ) as EndD (V ), σh ,
and we characterize the involutions which can be represented as adjoint involutions
with respect to a hyperbolic form. We de¬ne a slightly more general notion of
index which takes into account the Schur index of the algebra. A (weak) analogue
of Springer™s theorem on odd degree extensions is discussed in the ¬nal subsection.
6.A. Isotropic ideals. Let (A, σ) be a central simple algebra with involution
(of any kind) over a ¬eld F of arbitrary characteristic.
(6.1) De¬nition. For every right ideal I in A, the orthogonal ideal I ⊥ is de¬ned
by
I ⊥ = { x ∈ A | σ(x)y = 0 for y ∈ I }.
It is clearly a right ideal of A, which may alternately be de¬ned as the annihilator
of the left ideal σ(I):
I ⊥ = σ(I)0 .
(6.2) Proposition. Suppose the center of A is a ¬eld. For every right ideal I ‚ A,
rdim I + rdim I ⊥ = deg A and I ⊥⊥ = I. Moreover, if (A, σ) = EndD (V ), σh and
I = HomD (V, W ) for some subspace W ‚ V , then
I ⊥ = HomD (V, W ⊥ ).
Proof : Since rdim σ(I) = rdim I, the ¬rst relation follows from the corresponding
statement for annihilators (??). This ¬rst relation implies that rdim I ⊥⊥ = rdim I.
Since the inclusion I ‚ I ⊥⊥ is obvious, we get I = I ⊥⊥ . Finally, suppose I =
HomD (V, W ) for some subspace W ‚ V . For every f ∈ EndD (V ), g ∈ I we have
g(y) ∈ W and h f (x), g(y) = h x, σ(f ) —¦ g(y) for x, y ∈ V .
Therefore, σ(f ) —¦ g = 0 if and only if f (x) ∈ W ⊥ , hence
I ⊥ = HomD (V, W ⊥ ).


A similar result holds if (A, σ) = (E — E op , µ), where E is a central simple F -
algebra and µ is the exchange involution, although the reduced dimension of a right
op
ideal is not de¬ned in this case. Every right ideal I ‚ A has the form I = I1 — I2
where I1 (resp. I2 ) is a right (resp. left) ideal in E, and
op
(I1 — I2 )⊥ = I2 — (I1 )op .
0 0

Therefore, by (??),
dimF I ⊥ = dimF A ’ dimF I and I ⊥⊥ = I
for every right ideal I ‚ A.
In view of the proposition above, the following de¬nitions are natural:
(6.3) De¬nitions. Let (A, σ) be a central simple algebra with involution over a
¬eld F . A right ideal I ‚ A is called isotropic (with respect to the involution
σ) if I ‚ I ⊥ . This inclusion implies rdim I ¤ rdim I ⊥ , hence (??) shows that
rdim I ¤ 1 deg A for every isotropic right ideal.
2
§6. THE INDEX 73


The algebra with involution (A, σ)”or the involution σ itself”is called isotropic
if A contains a nonzero isotropic ideal.
If the center of A is a ¬eld, the index of the algebra with involution (A, σ) is
de¬ned as the set of reduced dimensions of isotropic right ideals:
ind(A, σ) = { rdim I | I ‚ I ⊥ }.
Since the (Schur) index of A divides the reduced dimension of every right ideal,
the index ind(A, σ) is a set of multiples of ind A. More precisely, if (A, σ)
EndD (V ), σh for some hermitian or skew-hermitian space (V, h) over a division
algebra D and w(V, h) denotes the Witt index of (V, h), then ind A = deg D and
ind(A, σ) = { deg D | 0 ¤ ¤ w(V, h) }
since (??) shows that the isotropic ideals of EndD (V ) are of the form HomD (V, W )
with W a totally isotropic subspace of V , and rdim HomD (V, W ) = dimD W deg D.
Thus, if ind(A, σ) contains at least two elements, the di¬erence between two con-
secutive integers in ind(A, σ) is ind A. If ind(A, σ) has only one element, then
ind(A, σ) = {0}, which means that (A, σ) is anisotropic; this is the case for in-
stance when A is a division algebra.
We extend the de¬nition of ind(A, σ) to the case where the center of A is
(E — E op , µ) for some central simple F -algebra E where µ
F — F ; then (A, σ)
is the exchange involution, and we de¬ne ind(A, σ) as the set of multiples of the
1
Schur index of E in the interval [0, 2 deg E]:
deg E
ind(A, σ) = ind E 0¤ ¤ .
2 ind E
(Note that deg E = deg(A, σ) is not necessarily even).
(6.4) Proposition. For every ¬eld extension L/F ,
ind(A, σ) ‚ ind(AL , σL ).
Proof : This is clear if the center of AL = A —F L is a ¬eld, since scalar extensions
preserve the reduced dimension of ideals and since isotropic ideals remain isotropic
under scalar extension. If the center of A is not a ¬eld, the proposition is also clear.
Suppose the center of A is a ¬eld K properly containing F and contained in L;
then by (??),
(A —K L) — (A —K L)op , µ .
(AL , σL )
Since the reduced dimension of every right ideal in A is a multiple of ind A and
since ind(A —K L) divides ind A, the reduced dimension of every isotropic ideal of
(A, σ) is a multiple of ind(A —K L). Moreover, the reduced dimension of isotropic
ideals is bounded by 1 deg A, hence ind(A, σ) ‚ ind(AL , σL ).
2

For central simple algebras with a quadratic pair, we de¬ne isotropic ideals by
a more restrictive condition.
(6.5) De¬nition. Let (σ, f ) be a quadratic pair on a central simple algebra A over
a ¬eld F . A right ideal I ‚ A is called isotropic with respect to the quadratic pair
(σ, f ) if the following two conditions hold:
(1) σ(x)y = 0 for all x, y ∈ I.
(2) f (x) = 0 for all x ∈ I © Sym(A, σ).
74 II. INVARIANTS OF INVOLUTIONS


The ¬rst condition means that I ‚ I ⊥ , hence isotropic ideals for the quadratic
pair (σ, f ) are also isotropic for the involution σ. Condition (??) implies that every
x ∈ I ©Sym(A, σ) satis¬es x2 = 0, hence also TrdA (x) = 0. If char F = 2, the map f
1
is the restriction of 2 TrdA to Sym(A, σ), hence condition (??) follows from (??).
Therefore, in this case the isotropic ideals for (σ, f ) are exactly the isotropic ideals
for σ.
The algebra with quadratic pair (A, σ, f )”or the quadratic pair (σ, f ) itself”is
called isotropic if A contains a nonzero isotropic ideal for the quadratic pair (σ, f ).
(6.6) Example. Let (V, q) be an even-dimensional quadratic space over a ¬eld F
of characteristic 2, and let (σq , fq ) be the corresponding quadratic pair on EndF (V )
(see (??)). A subspace W ‚ V is totally isotropic for q if and only if the right ideal
HomF (V, W ) ‚ EndF (V ) is isotropic for (σq , fq ).
Indeed, the standard identi¬cation •q (see (??)) identi¬es HomF (V, W ) with
W —F V . The elements in W — V which are invariant under the switch involution
(which corresponds to σq under •q ) are spanned by elements of the form w —w with
w ∈ W , and (??) shows that fq —¦ •q (w — w) = q(w). Therefore, the subspace W
is totally isotropic with respect to q if and only if fq vanishes on HomF (V, W ) ©
Sym EndF (V ), σq . Proposition (??) shows that this condition also implies that
HomF (V, W ) is isotropic with respect to the involution σq .
Mimicking (??), we de¬ne the index of a central simple algebra with quadratic
pair (A, σ, f ) as the set of reduced dimensions of isotropic ideals:
ind(A, σ, f ) = { rdim I | I is isotropic with respect to (σ, f ) }.
6.B. Hyperbolic involutions. Let E be a central simple algebra with invo-
lution θ (of any kind) and let U be a ¬nitely generated right E-module. As in (??),
we use the involution θ to endow the dual of U with a structure of right E-module
θ—
U . For » = ±1, de¬ne
h» : ( θ U — • U ) — ( θ U — • U ) ’ E
by
h» (θ • + x, θ ψ + y) = •(y) + »θ ψ(x)
for •, ψ ∈ U — and x, y ∈ U . Straightforward computations show that h1 (resp.
h’1 ) is a nonsingular hermitian (resp. alternating) form on θ U — • U with respect
to the involution θ on E. The hermitian or alternating module (θ U — • U, h» ) is
denoted H» (U ). A hermitian or alternating module (M, h) over (E, θ) is called
hyperbolic if it is isometric to some H» (U ).
The following proposition characterizes the adjoint involutions with respect to
hyperbolic forms:
(6.7) Proposition. Let (A, σ) be a central simple algebra with involution (of any
kind ) over a ¬eld F of arbitrary characteristic. Suppose the center K of A is a
¬eld. The following conditions are equivalent:
(1) for every central simple K-algebra E Brauer-equivalent to A and every involu-
tion θ on E such that θ|K = σ|K , every hermitian or skew-hermitian module (M, h)
over (E, θ) such that (A, σ) EndE (M ), σh is hyperbolic;
(2) there exists a central simple K-algebra E Brauer-equivalent to A, an involution
θ on E such that θ|K = σ|K and a hyperbolic hermitian or skew-hermitian module
(M, h) over (E, θ) such that (A, σ) EndE (M ), σh ;
§6. THE INDEX 75


(3) 1 deg A ∈ ind(A, σ) and further, if char F = 2, the involution σ is either sym-
2
plectic or unitary;
(4) there is an idempotent e ∈ A such that σ(e) = 1 ’ e.
Proof : (??) ’ (??) This is clear.
(??) ’ (??) In the hyperbolic module M = θ U — • U , the submodule U is
totally isotropic, hence the same argument as in (??) shows that the right ideal
HomE (M, U ) is isotropic in EndE (M ). Moreover, since rdim U = 1 rdim M , we
2
have rdim HomE (M, U ) = 2 deg EndE (M ). Therefore, (??) implies that 1 deg A ∈
1
2
ind(A, σ). If char F = 2, the hermitian form h+1 = h’1 is alternating, hence σ is
symplectic if the involution θ on E is of the ¬rst kind, and is unitary if θ is of the
second kind.
1
(??) ’ (??) Let I ‚ A be an isotropic ideal of reduced dimension 2 deg A.
By (??), there is an idempotent f ∈ A such that I = f A. Since I is isotropic, we
have σ(f )f = 0. We shall modify f into an idempotent e such that I = eA and
σ(e) = 1 ’ e.
The ¬rst step is to ¬nd u ∈ A such that σ(u) = 1 ’ u. If char F = 2, we
may choose u = 1/2; if char F = 2 and σ is symplectic, the existence of u follows
from (??); if char F = 2 and σ is unitary, we may choose u in the center K of A,
since K/F is a separable quadratic extension and the restriction of σ to K is the
nontrivial automorphism of K/F .
We next set e = f ’f uσ(f ) and proceed to show that this element satis¬es (??).
Since f 2 = f and σ(f )f = 0, it is clear that e is an idempotent and σ(e)e = 0.
Moreover, since σ(u) + u = 1,
eσ(e) = f σ(f ) ’ f uσ(f ) ’ f σ(u)σ(f ) = 0.
Therefore, e and σ(e) are orthogonal idempotents; it follows that e + σ(e) also is an
idempotent, and e + σ(e) A = eA • σ(e)A. To complete the proof of (??), observe
that e ∈ f A and f = ef ∈ eA, hence eA = f A = I. Since rdim I = 1 deg A, 2
1
it follows that dimF eA = dimF σ(e)A = 2 dimF A, hence e + σ(e) A = A and
therefore e + σ(e) = 1.
(??) ’ (??) Let E be a central simple K-algebra Brauer-equivalent to A and
θ be an involution on E such that θ|K = σ|K . Let also (M, h) be a hermitian or
skew-hermitian module over (E, θ) such that (A, σ) EndE (M ), σh . Viewing
this isomorphism as an identi¬cation, we may ¬nd for every idempotent e ∈ A a
pair of complementary submodules U = im e, W = ker e in M such that e is the
projection M ’ U parallel to W ; then 1 ’ e is the projection M ’ W parallel
to U and σ(e) is the projection M ’ W ⊥ parallel to U ⊥ . Therefore, if σ(e) = 1 ’ e

we have U = U ⊥ and W = W ⊥ . We then de¬ne an isomorphism W ’ θ U — by ’
mapping w ∈ W to θ •w where •w ∈ U — is de¬ned by •w (u) = h(w, u) for u ∈ U .

This isomorphism extends to an isometry M = W • U ’ H» (U ) where » = +1 if

h is hermitian and » = ’1 if h is skew-hermitian.
(6.8) De¬nition. A central simple algebra with involution (A, σ) over a ¬eld F ”
or the involution σ itself”is called hyperbolic if either the center of A is isomorphic
to F — F or the equivalent conditions of (??) hold. If the center is F — F , then
the idempotent e = (1, 0) satis¬es σ(e) = 1 ’ e; therefore, in all cases (A, σ) is
hyperbolic if and only if A contains an idempotent e such that σ(e) = 1 ’ e. This
condition is also equivalent to the existence of an isotropic ideal I of dimension
1
dimF I = 2 dimF A if char F = 2, but if char F = 2 the extra assumption that σ is
76 II. INVARIANTS OF INVOLUTIONS


symplectic or unitary is also needed. For instance, if A = M2 (F ) (with char F = 2)
1
and σ is the transpose involution, then 2 deg A ∈ ind(A, σ) since the right ideal
I = { x x | x, y ∈ F } is isotropic, but (A, σ) is not hyperbolic since σ is of
yy
orthogonal type.
Note that (A, σ) may be hyperbolic without A being split; indeed we may have
1 1
ind(A, σ) = {0, 2 deg A}, in which case the index of A is 2 deg A.
From any of the equivalent characterizations in (??), it is clear that hyper-
bolic involutions remain hyperbolic over arbitrary scalar extensions. Characteriza-
tion (??) readily shows that hyperbolic involutions are also preserved by transfer.
Explicitly, consider the situation of §??: Z/F is a ¬nite extension of ¬elds, E is
a central simple Z-algebra and T is a central simple F -algebra contained in E, so
that E = T —F C where C is the centralizer of T in E. Let θ be an involution on E
which preserves T and let s : E ’ T be an involution trace. Recall from (??) that
for every hermitian or skew-hermitian module (M, h) over (E, θ) there is a transfer
M, s— (h) which is a hermitian or skew-hermitian module over (T, θ).
(6.9) Proposition. If h is hyperbolic, then s— (h) is hyperbolic.
Proof : If h is hyperbolic, (??) yields an idempotent e ∈ EndE (M ) such that
σh (e) = 1 ’ e. By (??), the involution σs— (h) on EndT (M ) extends σh , hence
e also is an idempotent of EndT (M ) such that σs— (h) (e) = 1 ’ e. Therefore, s— (h)
is hyperbolic.

In the same spirit, we have the following transfer-type result:
(6.10) Corollary. Let (A, σ) be a central simple algebra with involution (of any
kind ) over a ¬eld F and let L/F be a ¬nite extension of ¬elds. Embed L ’
EndF (L) by mapping x ∈ L to multiplication by x, and let ν be an involution on
EndF (L) leaving the image of L elementwise invariant. If (AL , σL ) is hyperbolic,
then A —F EndF (L), σ — ν is hyperbolic.
Proof : The embedding L ’ EndF (L) induces an embedding
(AL , σL ) = (A —F L, σ — IdL ) ’ A —F EndF (L), σ — ν .
The same argument as in the proof of (??) applies. (Indeed, (??) may be regarded
as the special case of (??) where C = Z = L: see ??).

(6.11) Example. Interesting examples of hyperbolic involutions can be obtained
as follows: let (A, σ) be a central simple algebra with involution (of any kind) over a
¬eld F of characteristic di¬erent from 2 and let u ∈ Sym(A, σ)©A— be a symmetric
unit in A. De¬ne an involution νu on M2 (A) by
’σ(a21 )u’1
a11 a12 σ(a11 )
νu =
’uσ(a12 ) uσ(a22 )u’1
a21 a22
for a11 , a12 , a21 , a22 ∈ A, i.e.,
10
νu = Int —¦ (σ — t)
0 ’u
where t is the transpose involution on M2 (F ).
Claim. The involution νu is hyperbolic if and only if u = vσ(v) for some v ∈ A.
§6. THE INDEX 77


Proof : Let D be a division algebra Brauer-equivalent to A and let θ be an involution
on D of the same type as σ. We may identify (A, σ) = EndD (V ), σh for some
hermitian space (V, h) over (D, σ), by (??). De¬ne a hermitian form h on V by
h (x, y) = h u’1 (x), y = h x, u’1 (y) for x, y ∈ V .
Under the natural identi¬cation M2 (A) = EndD (V • V ), the involution νu is the
adjoint involution with respect to h⊥(’h ). The form h⊥(’h ) is hyperbolic if and
only if (V, h) is isometric to (V, h ). Therefore, by Proposition (??), νu is hyperbolic
if and only if (V, h) is isometric to (V, h ). This condition is also equivalent to the
existence of a ∈ A— such that h (x, y) = h a(x), a(y) for all x, y ∈ V ; in view of
the de¬nition of h , this relation holds if and only if u = a’1 σ(a’1 ).

The fact that νu is hyperbolic if u = vσ(v) for some v ∈ A can also be readily
1 1 ’1
proved by observing that the matrix e = 2 v v 1 is an idempotent such that
νu (e) = 1 ’ e.
Hyperbolic quadratic pairs. By mimicking characterization (??) of hyper-
bolic involutions, we may de¬ne hyperbolic quadratic pairs as follows:

(6.12) De¬nition. A quadratic pair (σ, f ) on a central simple algebra A of even
1
degree over a ¬eld F of arbitrary characteristic is called hyperbolic if 2 deg A ∈

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