The set of real symplectic matrices is defined as
$$\mathrm { Sp } _ { n } ( \mathbb { R } ) = \mathrm { Sp } _ { 2 m } ( \mathbb { R } ) = \left\{ M \in \mathcal { M } _ { n } ( \mathbb { R } ) \mid M ^ { \top } J M = J \right\}$$
where $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is a subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$, stable under transposition and containing the matrix $J$.