Expression of the Riemannian metric on the Siegel domain?

Question

I'm looking for proof that, for the complex Siegel domain in $\mathbb C^{n}$ defined by: $$\mathcal H_{n} = \{ z=(z_1,\dots,z_n) \in \mathbb C^{n} \mid \operatorname{Im}(z_{n}) > \sum_{j=1}^{n-1} |z_j|^2 \}$$ the Riemannian metric $ds^2_z$ is given by \begin{align*} ds^2_z =&\; \frac{1}{(\operatorname{Im}(z_{n}) - \sum_{i=1}^{n-1} |z_i|^2)^2} \bigg( (\operatorname{Im}(z_{n}) - \sum_{i=1}^{n-1} |z_i|^2) \sum_{j=1}^{n-1} d z_j \otimes d \bar{z}_j + \sum_{j,k=1}^{n-1} \bar{z}_j z_k d z_j \otimes d\bar{z}_k \\ &+ \frac{1}{2i} \sum_{j=1}^{n-1} \big( \bar{z}_j d z_j \otimes d\bar{z}_{n} -z_j d z_{n} \otimes d \bar{z}_j \big) + \frac{1}{4} d z_{n} \otimes d \bar{z}_{n} \bigg). \end{align*} Someone can give me the proof of the expression of $ds^2_z$ or a reference where I can find the demonstration. Thank you in advance