= Witt index
{c}
{title2=$i_W(q)=\max_{q|_U=0}\dim U$}
The <Witt index> of a nondegenerate <quadratic form> $q$ on a finite-dimensional <vector space> $V$ is the largest dimension of a <totally isotropic subspace>: a linear subspace $U$ for which $q(u)=0$ for every $u\in U$. In characteristic different from two, the <Witt decomposition> writes
$$
q\cong H^{\perp r}\perp q_{\mathrm{an}},
$$
where $H$ is a <hyperbolic plane of a quadratic form>, $q_{\mathrm{an}}$ is an <anisotropic quadratic form>, and $r$ is the Witt index. The anisotropic part has no nonzero vector on which the form vanishes.<Introduction to quadratic forms by Bill Casselman>[Proposition 2.6, Corollaries 2.7–2.8 and 6.8]{ref}
For the real form $x_1^2+x_2^2-x_3^2-x_4^2$, the subspace spanned by $(1,0,1,0)$ and $(0,1,0,1)$ is totally isotropic. Projection onto the two negative coordinates is injective on any totally isotropic subspace, giving the matching upper bound two. More generally, a real form of signature $(a,b)$ has Witt index $\min(a,b)$.
The base <field> matters: $x^2+y^2$ has index zero over $\mathbb R$ but index one over $\mathbb C$, using the line spanned by $(1,i)$. In characteristic two, vanishing of the associated polar form on $U$ alone is insufficient; the condition remains $q|_U=0$. Over a finite field, the two nondegenerate types in dimension $2m$ have Witt indices $m$ and $m-1$.<The geometry of orthogonal groups over finite fields by Akihiro Munemasa>[section 2, definition of Witt index and Theorem 2.11]{ref}
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