We consider a matrix $M \in \mathrm { Sp } _ { n } ( \mathbb { R } )$ and $S \in \mathcal { S } _ { n } ( \mathbb { R } )$ a symmetric symplectic matrix with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$. Justify that $S$ is invertible then show that the matrix $O$ defined by $O = M S ^ { - 1 }$ belongs to the group $\mathrm { OSp } _ { n } ( \mathbb { R } )$.