Proposition 1

Suppose that A is a field. Let F be a non-isotropic finite-dimensional subspace of E; then E is the direct sum of F and $F^0$.

Indeed, since the restriction $\Phi'$ of $\Phi$ to F is non-degenerate by assumption, the mapping $d_{\Phi'}$ from F into its dual $F^*$ associated on the right with $\Phi'$ is injective, hence bijective since F and $F^*$ are two spaces of the same finite dimension. Consequently, for every $y \in E$, there exists one and only one element $y_0$ of F such that $\Phi(x, y) = \Phi(x, y_0)$ for every $x \in F$, that is, $y - y_0 \in F^0$; this proves that E is the direct sum of F and $F^0$.