Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$

Then a group action $\rho : \text{SL}(2,\mathbb{C})\times \text{H}^{+}_{3}\rightarrow \text{H}^{+}_{3}$ seems to be defined via linear fractional transformations, i.e

$$\rho(M, z+wj) \mapsto M(z+wj) = \frac{\alpha(z+wj)+\beta}{\gamma (z+wj)+\delta} = z^{*}+w^{*} j \in \text{H}_{3}\tag{2}$$

Where $$z^{*}=\frac{(\alpha z+\beta)(\bar{\gamma} \bar{z}+\bar{\delta})+\alpha \bar{\gamma}w^{2}}{|\gamma z+\delta|^{2}+|\gamma|^{2} w^{2}} , \space\space w^{*}= \frac{w}{|\gamma z+\delta|^{2}+|\gamma|^{2} w^{2}}\tag{3}$$

Does this induce a group action $\rho : \text{SL}(2,\mathbb{C})\times \mathcal{L}^2(\text{H}^{+}_3)\rightarrow \mathcal{L}^2(\text{H}^{+}_3)$ through the following map?

$$\rho(M, f)(z+wj)\mapsto f(M^{-1}(z+wj))\tag{4}$$

If so, can we further define the induced Lie algebra action $d\rho : \mathfrak{sl}(2, \mathbb{C})\times \mathcal{L}^2(\text{H}^{+}_3)\rightarrow \mathcal{L}^2(\text{H}^{+}_3)$ through the following map?

$$\begin{align}d\rho(m, f)(z+wj)\mapsto \frac{d}{dt}\biggr|_{t = 0}\rho(e^{tm}, f)(z+wj) &= \frac{d}{dt}\biggr|_{t = 0}f(e^{-tm}(z+wj))\\ &= -\frac{\partial f}{\partial u}m(z+wj)\end{align}\tag{5}$$

Where $u = z+wj\in \text{H}^{+}_3$.