= Metabolic bilinear form
{title2=$L=L^\perp$}
A metabolic bilinear form is a <nondegenerate> <symmetric bilinear form> $B$ on a finite-dimensional <vector space> $V$ possessing a <vector subspace> $L$ with
$$
L=L^\perp,\qquad L^\perp=\{v\in V:B(v,\ell)=0\text{ for every }\ell\in L\}.
$$
Such an $L$ is a <metabolizer>. Equivalently, it is a totally isotropic <vector subspace> of half the <dimension>: $B$ vanishes on $L\times L$ and $\dim V=2\dim L$. Nondegeneracy gives $\dim L^\perp=\dim V-\dim L$, proving the equivalence.
The <symmetric matrix> $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ is metabolic: the line spanned by $(1,0)$ is its own <bilinear orthogonal complement>. Over a <field> of <characteristic> different from two, every metabolic <symmetric bilinear form> is a sum of such hyperbolic planes. In <characteristic> two, metabolic need not mean alternating: over $\mathbb F_2$, the identity matrix of size two has metabolizer spanned by $(1,1)$ but has $B((1,0),(1,0))=1$.
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