Let $M \in \mathcal{S}_{2n}(\mathbb{R}) \cap \mathrm{Sp}_{2n}(\mathbb{R})$. In this question $\lambda = 1$. Show that $E_{1}$ has even dimension and that there exists a basis of $E_{1}$ that is orthonormal and of the form $(X_{1}, \ldots, X_{p}, J_{n} X_{1}, \ldots, J_{n} X_{p})$ where $2p$ is the dimension of $E_{1}$.