1. QUADRATIC FORMS AND ASSOCIATIVE ALGEBRAS 11

subspace Z just defined for {ei, f \ } . Applying our induction to Z and its subspace

H2T=2 Fei, w e obtain our lemma.

In the setting of the above lemma we call the expression of (1.2a) with 0

m G Z a weak W i t t decompositio n (of V with respect to p); we call it a

W i t t decompositio n if the restriction of p to Z is anisotropic. In particular, if

Z = {0}, we call it a split W i t t decomposition . If (1.2a) is a Witt decomposition,

we call Z a core subspace of V with respect to p, and call dim(Z) the core

dimensio n of (V, p), or simply, of p.

L e m m a 1.4. Suppose p is nondegenerate on V; then the following assertions

hold:

(i) V has a Witt decomposition.

(ii) If (1.2a) is a Witt decomposition and ( is the restriction of ip to Z, then

the integer m and the isomorphism class of (Z, Q are completely determined by

the isomorphism class of'(V, p).

(iii) Every totally isotropic subspace of V is contained in a totally isotropic

subspace of V of dimension m with the integer m determined as in (ii).

P R O O F . There is no problem if p is anisotropic. Suppose V has a nontrivial

totally isotropic subspace X. Clearly X is contained in a maximal totally isotropic

subspace / of V. By Lemma 1.3, we can find a weak Witt decomposition (1.2a) with

/ in place of Y^tLi Fz%- If Z contains a nonzero vector g such that p[g] = 0, then

Fg + I is totally isotropic, which is a contradiction as / is maximal. Therefore p is

anisotropic on Z, so that (1.2a) is a Witt decomposition. This proves (i). Let V =

Z' + ^2^=i(Fe[ + Ffl) be another Witt decomposition; suppose n m. Then, by

Theorem 1.2, we can find an F-linear bijection / of Z onto Z'^Y^lZT(Fei + Ffi)

such that p[xf] = p[x] for every x G Z. Since p is anisotropic on Z, we have

m = n, and at the same time we obtain (ii). Assertion (iii) is also proved, as we

have seen that X C I.

L e m m a 1.5. (i) Given (V, p) with nondegenerate /?, let a and ft be F-linear

injections of a vector space W into V such that p[xa\ = p[xft] for every x G W.

Suppose that p is nondegenerate on Wot. Then there exists an element 7 of O^

such that « 7 = ft. Moreover, such a 7 can be taken from SO^ if dim(F)

dim(VT).

(ii) Given (V, p) as in (i), let h and k be nonzero elements of V such that

p[h] = p[k}. Then there exists an element 7 of O^ such that hj = k. Moreover

such a 7 can be taken from SO^ if p[h] 7^ 0 and dim(V) 1, or if p[h] = 0 and

dim(V) 2.

P R O O F . We can find an F-linear bijection £ of Wa to Wft by (xa)£ = xft

for x G W. Then p[y£] — p[y] for every y G Wa. Applying Theorem 1.2 to

V = (Wa) + (Wa)± = (Wft) + (Wft)±1 we can extend £ to an element 7 of 0*\

Then aj = ft. If dim(V) dim(W), we can modify 7 by a suitable element of

O^^WaY) so that det(7) = 1. This proves (i).

As for (ii), suppose p[h] j^ 0; define a, ft : F — » V by xa = xh and xft = xk

for x G F. Then our assertion follows from (i). Suppose p[h] — 0. Then, by Lemma

1.3, V has weak Witt decompositions V = X + Fh + Fg = Y + Fk + Fi with