Metabolic bilinear form ($L=L^{\perp }$)

A metabolic bilinear form is a nondegenerate symmetric bilinear form $B$ on a finite-dimensional vector space $V$ possessing a vector subspace $L$ withSuch 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 $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ 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 $B((1,0),(1,0))=1$.