I am working on a project that I suspect requires calculations involving Möbius transformations on 3-dimensional space $\mathbb{R}^3$, identified with the quaternions $\mathbb{H}$ with $k$-component zero (in other words, the action of 4-dimensional hyperbolic isometries on the 3-dimensional boundary of $H^4$). My understanding is that such transformations can be identified with $2 \times 2$ quaternionic matrices $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ satisfying $ab^*,cd^*,c^*a,d^*b \in \mathbb{R}^3$ and $ad^*-bc^*=1$.
I would like a proof that this is the correct group. More importantly, I would like to check that the usual nice properties carry over from the real and complex cases (any Möbius transformation is fully determined by its action on 3 points, it can be decomposed as a composition of simple transforms, etc.), and what things need to be adjusted. If anyone knows a good reference for these sort of questions, it would probably save me a lot of time, and I would be most grateful.