The isometry group of $n$-dimensional Einstein gyrogroup

The space of $n$-dimensional relativistic velocities normalized to $c=1$, $$\mathbb{B}=\{\mathbf{v}\in {\mathbb{R}}^n\text{\hspace{0.17em}⁣}:\mathrm{\parallel }\mathbf{v}\mathrm{\parallel }<1\},$$ is naturally associated with Einstein velocity addition ${\oplus }_E$, which induces the rapidity metric $d_E$ on $\mathbb{B}$ given by $d_E($\mathbf{u}$, $\mathbf{v}$) = \tanh^{-1}|-$\mathbf{u}$\oplus_E$\mathbf{v}$|$. This metric is also known as the Cayley-Klein metric. We give a complete description of the isometry group of $(\mathbb{B},d_E)$, along with its composition law.