Any element of the symplectic group $\operatorname{Sp}(2n,\mathbb{R})$ can be decomposed using the Euler decomposition into the product of three matrices. \begin{equation} S = O\begin{pmatrix}D & 0 \\ 0 & D^{-1}\end{pmatrix}O' \quad \quad \forall S \in Sp(2n,\mathbb{R}), \end{equation} where $O, O'$ are orthogonal and symplectic - $\left(\operatorname{Sp}(2n,\mathbb{R})\cap \operatorname{O}(2n) \cong U(n)\right)$; $D$ is positive definite and diagonal.
Does this matrix decomposition correspond to some Lie group decomposition or group manifold decomposition i.e. Cartesian product of submanifolds? In other words, what more does this tell me about the group itself and its manifold?