The real symplectic group manifold is diffeomorphic to this Cartesian product of manifolds: \begin{equation} \operatorname{Sp}(2n,\mathbb{R}) \simeq \operatorname{U}(n) \times \mathbb{R}^{n(n+1)}. \end{equation} from Arnol'd and Givental's 'Symplectic Geometry'.
EDIT 1: Is there a group isomorphism that fits this diffeomorphic breakdown? i.e. a decomposition of $\operatorname{Sp}(2n,\mathbb{R})$ into $\operatorname{U}(n)$ and something else.
EDIT 2: With its lack of normal subgroups, by decomposition I mean a more general product like Zappa-Szép.