Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Show that there exists a non-zero natural integer $q$ and vector subspaces of $\mathcal{M}_{2n,1}(\mathbb{R})$, denoted $F_{1}, \ldots, F_{q}$ such that
[(e)] $\forall i \in \{1,\ldots,q\}$, $\dim F_{i} \in \{2,4\}$;
[(f)] $\forall i \in \{1,\ldots,q\}$, the matrix of the application $m_{F_{i}}$ induced by $m$ on $F_{i}$ in a certain basis is of the form $$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$
Let $M \in \mathcal{A}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$, and let $m$ be the linear map canonically associated with $M$. Show that there exists a non-zero natural integer $q$ and vector subspaces of $\mathcal{M}_{2n,1}(\mathbb{R})$, denoted $F_{1}, \ldots, F_{q}$ such that
\begin{itemize}
\item[(a)] $F_{1} \oplus \cdots \oplus F_{q} = \mathcal{M}_{2n,1}(\mathbb{R})$;
\item[(b)] $\forall i \in \{1,\ldots,q\}$, $F_{i}$ is stable under $M$ and under $J_{n}$;
\item[(c)] $\forall i \in \{1,\ldots,q\}$, $F_{i}^{\perp}$ is stable under $M$ and under $J_{n}$;
\item[(d)] $\forall (i,j) \in \{1,\ldots,q\}^{2}$, $i \neq j \Longrightarrow \forall (Y,Z) \in F_{i} \times F_{j}$, $\langle Y,Z \rangle = 0 = \varphi(Y,Z)$;
\item[(e)] $\forall i \in \{1,\ldots,q\}$, $\dim F_{i} \in \{2,4\}$;
\item[(f)] $\forall i \in \{1,\ldots,q\}$, the matrix of the application $m_{F_{i}}$ induced by $m$ on $F_{i}$ in a certain basis is of the form
$$J_{1} \quad \text{or} \quad \left(\begin{array}{cc} \sqrt{-\lambda} J_{1} & 0_{2,2} \\ 0_{2,2} & \frac{1}{\sqrt{-\lambda}} J_{1} \end{array}\right).$$
\end{itemize}