Consider the group $\operatorname{PSL}(2,C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is isometric with respect to the hyperbolic metric. To prove this you write each Möbius transformation as a product of inversions in spheres and show that each of these act by isometries.

My question: can anyone point out a reference where this is done rigorously and explicitly (or explain the computation)? Everywhere I read the proof that inversions in spheres act by isometries on the unit ball with the hyperbolic metric is left as an "exercise" which I cannot do. I am teaching a class on this but need to understand myself...