Orthosymplectic diagonalization in Williamson's theorem

In this paper, we provide an algebraic condition on any $2n \times 2n$ real symmetric positive definite matrix which is necessary and sufficient for the matrix to be diagonalized by an orthosymplectic matrix in the sense of Williamson's theorem.
Comment: This work is going to become part of a bigger project that would contain wider applications of the main result presented in the paper