Geometry of the slice regular M\"obius transformations of the quaternionic unit ball

For the quaternionic unit ball $\mathbb{B}$, let us denote by $\mathcal{M}(\mathbb{B})$ the set of slice regular M\"obius transformations mapping $\mathbb{B}$ onto itself. We introduce a smooth manifold structure on $\mathcal{M}(\mathbb{B})$, for which the evaluation(-action) map of $\mathcal{M}(\mathbb{B})$ on $\mathbb{B}$ is smooth. The manifold structure considered on $\mathcal{M}(\mathbb{B})$ is obtained by realizing this set as a quotient of the Lie group $\mathrm{Sp}(1,1)$, Furthermore, it turns out that $\mathbb{B}$ is a quotient as well of both $\mathcal{M}(\mathbb{B})$ and $\mathrm{Sp}(1,1)$. These quotients are in the sense of principal fiber bundles. The manifold $\mathcal{M}(\mathbb{B})$ is diffeomorphic to $\mathbb{R}^4 \times S^3$.