I want to show that the groups $Sp(2,R)$ and $SU(1,1)$ are isomorphic. But more generally I wonder, if two groups $G_1, G_2$ leave the matrices $M_1, M_2 $ invariant respectively. Then can I show an isomorphism between the groups if I show $$SM_1S^{\dagger}=M_2.$$ Where $S$ is a unitary matrix.
More specifically $Sp(2,R)$ and $SU(1,1)$ leave, $$ \begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}, \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}, $$ invariant, respectively. This is what I have been led to believe but I am not sure why it would be true. Is it? Or is there an easier alternative way of showing isomorphism in this case?