On the generators of \(\text{Sp}_ n(\mathbb{Z})\) (Q677147)

Let \(\mathbb{Z}\) denote the integers. \textit{E. Witt} [Abh. Math. Semin. Univ. Hamb. 14, 323-337 (1941; Zbl 0025.01701)] showed that the group \(\text{Sp}_n(\mathbb{Z})\) is generated by the matrices \(\bigl(\begin{smallmatrix} 1&S\\ 0 &1\end{smallmatrix}\bigr)\) (\(S\in M_n(\mathbb{Z})\), \(S=S^t\)), \(\bigl(\begin{smallmatrix} U^t&0\\ 0 &U^{-1}\end{smallmatrix}\bigr)\) (\(U\in\text{GL}_n(\mathbb{Z})\)), and \(\bigl(\begin{smallmatrix} 0_n &1_n\\ -1_n &0_n\end{smallmatrix}\bigr)\). The author gives a short proof for this theorem.