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Let $n$ be an integer $\ge 1$. We define the $2n\times 2n$ matrix $\sigma$ with $n\times n$ blocks by $$ \sigma=\begin{pmatrix}0&I_n \\-I_n&0\end{pmatrix}. $$ The symplectic group $Sp(n)$ is defined as the set of $2n\times 2n$ matrices $S$ such that $ S^*\sigma S=\sigma. $ It is a classical fact that the symplectic group is a subgroup of $Sl(2n, \mathbb R)$ and is generated by matrices $$ \begin{pmatrix}B^{-1}&0 \\0&B^*\end{pmatrix}\text{(with $B\in Gl(n,\mathbb R)$}),\quad \begin{pmatrix}I_n&0 \\A&I_n\end{pmatrix}\text{(with $A$ symmetric $n\times n$}) \tag{$\ast$}$$ and linear mapping $M_j, 1\le j\le n$ such that $ (x_j, \xi_j)\mapsto (\xi_j, -x_j),\ \text {other coordinates fixed}. $ I claim that it is enough to get a set of generators to consider ($\ast$) along with $$ \begin{pmatrix}I_n&C \\0&I_n\end{pmatrix}\text{(with $C$ symmetric $n\times n$ }), \text{as well as $\sigma$}. $$ It should be classical. I would be grateful for a reference in the literature.

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