Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1, f_1 \in E$ with $\omega(e_1, f_1) = 1$, and let $\delta_1$ be a composition of at most two symplectic transvections such that $\delta_1(u(e_1)) = e_1$. Let $\tilde { f } _ { 1 }$ denote the vector $\delta _ { 1 } \left( u \left( f _ { 1 } \right) \right)$. Show that there exists a composition $\delta _ { 2 }$ of at most two symplectic transvections of $E$ such that $$\left\{ \begin{array} { l }
\delta _ { 2 } \left( e _ { 1 } \right) = e _ { 1 } \\
\delta _ { 2 } \left( \tilde { f } _ { 1 } \right) = f _ { 1 }
\end{array} \right.$$ One may adapt the proof of the preceding lemma.
Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$ and let $u \in \operatorname { Symp } _ { \omega } ( E )$. Let $e_1, f_1 \in E$ with $\omega(e_1, f_1) = 1$, and let $\delta_1$ be a composition of at most two symplectic transvections such that $\delta_1(u(e_1)) = e_1$. Let $\tilde { f } _ { 1 }$ denote the vector $\delta _ { 1 } \left( u \left( f _ { 1 } \right) \right)$. Show that there exists a composition $\delta _ { 2 }$ of at most two symplectic transvections of $E$ such that
$$\left\{ \begin{array} { l }
\delta _ { 2 } \left( e _ { 1 } \right) = e _ { 1 } \\
\delta _ { 2 } \left( \tilde { f } _ { 1 } \right) = f _ { 1 }
\end{array} \right.$$
One may adapt the proof of the preceding lemma.