 Hyperbolic extension of a Möbius transformation

A Möbius transformation of the ideal two-sphere extends uniquely to an orientation-preserving isometry of three-dimensional hyperbolic space. In upper half-space, translations, complex similarities, and $z\mapsto -1/z$ extend respectively to $(z,t)\mapsto (z+b,t)$, $(\lambda z,|\lambda |t)$, and $(-\overline{z},t)/(|z{|}^2+t^2)$. Each preserves $(|dz{|}^2+d{t}^2)/t^2$; composing these extensions constructs the general case. Ball-origin stabilizers are ordinary rotations, represented on the boundary by $SU(2)$ modulo its central signs.