We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$, and let $u \in \mathrm{GL}(E)$ be the unique automorphism such that $\omega_1(x,y) = \omega(u(x),y)$ for all $(x,y) \in E^2$. We assume that $E$ is of dimension 4 and that $u$ has no real eigenvalue. Let $\widetilde{\mathcal{B}} = (z_1, z_2, y_1, -y_2)$ be the basis constructed in questions 16--18, where $\operatorname{Mat}_{\widetilde{\mathcal{B}}}(\omega) = J_4$.
Show that there exist $r > 0$ and $\theta \in \mathbb { R } \backslash \pi \mathbb { Z }$ such that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( u ) = r \left( \begin{array} { c c } R _ { \theta } & 0 \\ 0 & R _ { - \theta } \end{array} \right)$$ where $R _ { \theta } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right)$, and conclude that $$\operatorname { Mat } _ { \widetilde { \mathcal { B } } } ( \omega ) = J _ { 4 } \quad \text{and} \quad \operatorname { Mat } _ { \widetilde { \mathcal { B } } } \left( \omega _ { 1 } \right) = r \left( \begin{array} { c c } 0 & - R _ { - \theta } \\ R _ { \theta } & 0 \end{array} \right).$$