be the Siegel upper half-space, with boundary CR structure induced by its embedding in $\mathbb C^{n+1}$. Define the Cayley transform $C:B^{n+1}\to\mathcal U^n$ by
and under the standard identification of $\mathbb H^n$ with $\mathbb C^n\times\mathbb R$, the map $\Theta^{-1}\circ C_{\partial}$ identifies the punctured sphere $\partial B^{n+1}\setminus\{(0,\dots,0,-1)\}$ with $\mathbb H^n$ as a CR manifold.