We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Show that there exists a unique $u \in \mathrm { GL } ( E )$ such that $\omega _ { 1 } ( x , y ) = \omega ( u ( x ) , y )$ for all $( x , y ) \in E ^ { 2 }$. Show then that $u$ belongs to the set $\mathcal { S }$ defined by
$$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$