I know non-identity 2 elements in $PSL(2,\mathbb{R})$ commutes iff they have the same fixed points in $\hat{\mathbb{C}}$.
But for $PSL(2,\mathbb{C})=Isom^{+}(\mathbb{H}^3)$, it seems a bit tricky for the iff conclusion.(Sharing fixed points in $\hat{\mathbb{R}^3}$+ some condition)
For $A$ being parabolic or hyperbolic, it seems same conclusion that $B$ also is parabolic or hyperbolic preserving same fixed points.
For $A$ being elliptic, $B$ also must preserve the same fixe point with $A$ and there is some extra condition I think. It seems related to under what condition two elements in $SO(3)$ commutes?
Any Guru can shine some light on this? Thank you very much.