For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form. Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$. a) Show that $G$ is non-singular. b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:
For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.
For the rest of this problem, we assume that $\varphi$ is a non-degenerate symmetric bilinear form on $E$, and we denote by $q$ its quadratic form.
Let $F$ be a singular vector subspace of $E$. We assume that $(e_1, \ldots, e_s)$ is a basis of $F \cap F^\perp$. We denote by $G$ a supplementary subspace of $F \cap F^\perp$ in $F$.
a) Show that $G$ is non-singular.
b) Demonstrate by induction on the dimension of $F \cap F^\perp$ (starting with $\operatorname{dim}(F \cap F^\perp) = 1$, then $\operatorname{dim}(F \cap F^\perp) > 1$) that there exist $s$ planes $P_1, \ldots, P_s$ of $E$ such that the following three properties are verified:
\begin{enumerate}
\item For all $i \in \{1,\ldots,s\}$, $(P_i, q_{/P_i})$ is an artinian plane containing $e_i$.
\item For all $(i,j) \in \{1,\ldots,s\}^2$ with $i \neq j$, $P_i$ is orthogonal to $P_j$.
\item For all $i \in \{1,\ldots,s\}$, $P_i$ is orthogonal to $G$.
\end{enumerate}