For some matrix $M$ of 2x2 block entries with diagonals
$$M_{jj} = \begin{bmatrix}0&a\\a&0\end{bmatrix},$$
where $a$ is a real constant, and with off diagonal elements
$$M_{jk} = \begin{bmatrix}0&b_{jk}\\b_{jk}^*&0\end{bmatrix},$$
where $j, k$ are integers ranging between 1 and $n$. Hence the dimensions of $M$ is $2n$x$2n$. How can I find the transforming simplectic matrix $S$ which relates $M$ to the block diagonal matrix $N$, where the diagonal elements are
$$N_{jj} = \begin{bmatrix}0&c_j\\c_j&0\end{bmatrix}.$$ Specifically, how can I find $S$ such that
$$M = S^\top N S$$