How do I derive the center of the symplectic group?
I define the real symplectic group to be the set of $2n \times 2n$ matrices satisfying \begin{equation} S\Omega S^\intercal = \Omega, \end{equation} where \begin{equation} \Omega = \bigoplus_{i=1}^n \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}. \end{equation}
The center of a group is the set of matrices that commute with every one of its elements. I have read that the center of the real symplectic group is $\{I_{2n},-I_{2n}\}$. How do I prove this?
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