Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname{Vect}(e_1, f_1)$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. Show that the restriction $\omega _ { P ^ { \omega } }$ of $\omega$ to $P ^ { \omega } \times P ^ { \omega }$ equips $P ^ { \omega }$ with a symplectic space structure and that the endomorphism $v _ { P ^ { \omega } }$ induced by $v$ on $P ^ { \omega }$ is a symplectic endomorphism.
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Let $P = \operatorname{Vect}(e_1, f_1)$ with $\omega(e_1,f_1)=1$, and $v = \delta \circ u$ where $u \in \operatorname{Symp}_{\omega}(E)$ and $\delta$ is a composition of symplectic transvections with $v|_P = \mathrm{id}_P$. Show that the restriction $\omega _ { P ^ { \omega } }$ of $\omega$ to $P ^ { \omega } \times P ^ { \omega }$ equips $P ^ { \omega }$ with a symplectic space structure and that the endomorphism $v _ { P ^ { \omega } }$ induced by $v$ on $P ^ { \omega }$ is a symplectic endomorphism.