Diffeomorphism symplectic group

Question

I want to show that $\text{Sp}(2, \mathbb{R})$ is diffeomorphic to $S^1 \times \mathbb{R}^2$.

So far I have that $\text{Sp}(2, \mathbb{R}) \cong \text{SL}(2, \mathbb{R})$. Is that correct?

Answer 1

Yes that is correct. To continue I might be good to look at the action of $SO(2)$ let's say from the left on $Sp(2,\mathbb{R})$. Can you find an element in $SO(2)$ such that the resulting matrix is of the form $\begin{pmatrix} e&f\\0&1/e\end{pmatrix}$?(Use $(a,c)\ne(0,0)$ and that elements in $SO(2)$ are of the form $\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{pmatrix}$)