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On page 75 of Eichler and Zagier's Theory of Jacobi Forms, it is claimed that $SP(4;\mathbb{Z})$ is generated by matrices of the form $$\begin{pmatrix} a & 0 & b & 0 \\ 0 & 1 & 0 & 0 \\ c & 0 & d & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \quad \mathrm{where}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2;\mathbb{Z})$$ and $$\begin{pmatrix} 1 & 0 & 0 & \mu \\ \lambda & 1 & \mu& 0 \\ 0 & 0 & 1 & -\lambda \\ 0 & 0 & 0 & 1\end{pmatrix}, \quad \mathrm{where} \; \lambda,\mu \in \mathbb{Z}$$ (i.e. the Jacobi group) together with the matrix $$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 &0 \end{pmatrix}.$$

I don't find it obvious how to get the usual generators of $SP(4;\mathbb{Z})$ (for example, the generators found by Hua and Reiner, or even the matrices $K$ and $L$ here ) from these, and if it is indeed well-known then I would like to cite a reference for it. Thanks for any help.

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  • $\begingroup$ I don't exactly know what happened over integers, but over a local field, these matrices are enough to generate the whole $Sp_4$. Let $\alpha$ be the short root and $\beta$ be the long root. In the language of Chevelley basis, the first matrix contains elements of the form $X_\beta(r), w_\beta$, the second matrix contains elements of the form $X_\alpha(\lambda), X_{\alpha+\beta}(\mu)$ and the third matrix is $w_\alpha$. Since the Weyl group is generated by $w_\alpha, w_\beta$, and the maximal unipotent group of a Borel subgroup can be generated by $X_\alpha, X_\beta, X_{\alpha+\beta}$ $\endgroup$
    – Q. Zhang
    Aug 24, 2016 at 23:29
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    $\begingroup$ The argument of Qing Z shows that your group in question is generated by the unipotent elements $X_{\alpha}(\mathbb{Z})$ for $all$ (positive or negative) roots $\alpha$. Thus you need only show that $SP_4(\mathbb{Z})$ is generated by unipotent elements. This is stated in the Bass-Milnor-Serre paper on the congruence subgroup problem (I don't have the precise place where they state it, right now). $\endgroup$
    – Venkataramana
    Aug 25, 2016 at 2:14

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