= Witt decomposition
{c}
{title2=$q\cong H^{\perp i_W(q)}\perp q_{\mathrm{an}}$}
For a finite-dimensional nonsingular <quadratic form> $q$ over a <field> of characteristic different from two, a <Witt decomposition> is an orthogonal splitting
$$
q\cong H^{\perp r}\perp q_{\mathrm{an}},
$$
where $H(u,v)=uv$ is a <hyperbolic plane of a quadratic form> and $q_{\mathrm{an}}$ is an <anisotropic quadratic form>. The integer $r$ is the <Witt index>. Both $r$ and the isometry class of the anisotropic part are determined by $q$, although the particular subspaces realizing the splitting need not be unique.<Introduction to quadratic forms by Bill Casselman>[Proposition 2.6, Corollaries 2.7–2.8 and 6.8]{ref}
Over $\mathbb Q$, take $q(x,y,z)=x^2+y^2-z^2$. Set $x=(u+v)/2$, $z=(u-v)/2$ and $y=w$. Then
$$
q=uv+w^2,
$$
so $q\cong H\perp\langle1\rangle$. The one-dimensional remainder is anisotropic and the Witt index is one. This is an isometry of quadratic spaces, stronger than merely expressing the underlying vector space as a direct sum.
The construction starts from an isotropic vector, finds a partner with nonzero polar pairing, and splits off the hyperbolic plane they generate. Repeating the process ends when no nonzero isotropic vector remains. The decomposition makes the isotropic directions and the residual arithmetic obstruction visible separately.
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