Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with $$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$ what known decompositions exist such that the matrix $S$ is decomposed into $S=ABC$ whereby $A,B,C \in \text{Sp}(2n,\mathbb Z)$?
When $S'\in\text{Sp}(2n,\mathbb R)$, there are plenty of decompositions (Euler, Iwasawa, eigenvalue decomposition) whereby $S'=ABC$ with $A,B,C \in\text{Sp}(2n,\mathbb R)$ but as far as I can tell there is no reason that these decompositions of $S\in\text{Sp}(2n,\mathbb Z)$ will produces matrices $A,B,C\in \text{Sp}(2n,\mathbb Z)$ with integer elements.
On the other hand, using Smith decomposition or Hermite decomposition, there is no reason to suggest that the resulting matrices will be symplectic, even if they are all integer.
This question is also related to this question, except for $\mathbb Z$ rather than $\mathbb Z/(d \mathbb Z)$.