Let $g, h$ be non-identity Möbius transformations. Prove that $g$ and $h$ commute if and only if one of the following holds:

1. $\operatorname{Fix}(g)=\operatorname{Fix}(h)$;

2. $g, h$ are involutions each of which exchanges the other's fixed points.

Give an example to show that the second case can occur.