Explain briefly how to extend a Möbius transformation

$$T: z \mapsto \frac{a z+b}{c z+d} \quad \text { with } a d-b c=1$$

from the boundary of the upper half-space $\mathbb{R}_{+}^{3}$ to give a hyperbolic isometry $\widetilde{T}$of the upper half-space. Write down explicitly the extension of the transformation $z \mapsto \lambda^{2} z$ for any constant $\lambda \in \mathbb{C} \backslash\{0\}$.

Show that, if $\tilde{T}$ has an axis, which is a hyperbolic line that is mapped onto itself by $\tilde{T}$ with the orientation preserved, then $\widetilde{T}$moves each point of this axis by the same hyperbolic distance, $\ell$ say. Prove that

$$\ell=2|\log | \frac{1}{2}\left(a+d+\sqrt{(a+d)^{2}-4}\right)||$$