Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ where $Z$ is symmetric $n*n$ matrix, the siegel upper half plane model is $\mathbb{Im}\, W>0$ where $W$ is $n*n$ matrix.Then the Cayley transformation is $W=i(I+Z)(I-Z)^{-1}$. What about type 1, type 3 domains?
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