Proving $O(1,1)$ is diffeomorphic to the disjoint union of 4 copies of the real line.

Question

I'm trying to prove that the pseudo-orthogonal group $O(1,1)$ is diffeomorphic to the disjoint union of four copies of $\mathbb{R}$. Firstly I tried to solve the equation $AI_{p,q}A^T = I_{p,q}$ where $I_{p,q} = diag(\underbrace{1, \dots, 1}_{p},\underbrace{-1,\dots,-1}_{q})$ and $A \in O(p,q)$ . I obtained that each matrix in $O(p,q)$ is of the form $ \left( \begin{array}{cc} coshx & sinhx \\ sinhx & coshx \end{array} \right) $, $ \left( \begin{array}{cc} -coshx & sinhx \\ sinhx & -coshx \end{array} \right) $, $ \left( \begin{array}{cc} coshx & sinhx \\ -sinhx & -coshx \end{array} \right) $ or $ \left( \begin{array}{cc} -coshx & sinhx \\ -sinhx & coshx \end{array} \right) $. I'm struggling with what to do now. I think I want to show that each of these matrices are diffeomorphic to a copy of the real line but I'm not too sure how to go about doing this.