The CR sphere $\mathbb{S}^{2 n+1}$ is the boundary of the unit ball $B$ of $\mathbb{C}^{n+1}$. In coordinates, $\zeta=\left(\zeta_1, \ldots, \zeta_{n+1}\right) \in \mathbb{S}^{2 n+1}$ if and only if $\zeta \cdot \bar{\zeta}=\sum_{j=1}^{n+1}\left|\zeta_j\right|^2=1$.
The Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$ and the sphere are equivalent via the Cayley transform $\mathcal{C}: \mathbb{H}^n \rightarrow \mathbb{S}^{2 n+1} \backslash(0,0, \ldots, 0,-1)$ given by$$\mathcal{C}(z, t)=\left(\frac{2 z}{1+|z|^2+i t}, \frac{1-|z|^2-i t}{1+|z|^2+i t}\right)$$and with inverse$$\mathcal{C}^{-1}(\zeta)=\left(\frac{\zeta_1}{1+\zeta_{n+1}}, \ldots, \frac{\zeta_n}{1+\zeta_{n+1}}, \operatorname{Im} \frac{1-\zeta_{n+1}}{1+\zeta_{n+1}}\right)$$The Jacobian determinant (really a volume density) of this transformation is given by$$\left|J_{\mathcal{C}}(z, t)\right|=\frac{2^{2 n+1}}{\left(\left(1+|z|^2\right)^2+t^2\right)^{n+1}}$$My questions are how to prove these identities $\mathcal{C}(z, t)$, $\mathcal{C}^{-1}(\zeta)$ and $\left|J_{\mathcal{C}}(z, t)\right|$? Welcome for any comments and help!