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 $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $z_1, z_2, y_1, y_2 \in E$ be as defined in question 16. Show that $$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$
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 $\mathcal{B}$ be a basis of $E$ such that $\operatorname{Mat}_{\mathcal{B}}(\omega) = J_4$, and let $z_1, z_2, y_1, y_2 \in E$ be as defined in question 16.
Show that
$$\begin{aligned} & \omega \left( z _ { 1 } , z _ { 2 } \right) = \omega \left( y _ { 1 } , y _ { 2 } \right) = 0 \\ & \omega \left( z _ { 1 } , y _ { 1 } \right) = - \omega \left( z _ { 2 } , y _ { 2 } \right) \\ & \omega \left( z _ { 1 } , y _ { 2 } \right) = \omega \left( z _ { 2 } , y _ { 1 } \right) \end{aligned}$$