Notation: $GL(n, \mathbb{Z})$ to be the set of  $n \times n$ invertible matrix, and $M_{m \times n}(\mathbb{Z})$ be the set of $m \times n$ matrix with integer entries.

I have the following conjecture:
Given $n \geq 2$ and two matrices $W_4$ and $S_2$ satisfy
\begin{equation}
    W_4 \in GL(n, \mathbb{Z}), S_2 \in GL(n, \mathbb{Q})
\end{equation}
Moreover $S_2^{-1} \in GL(n, \mathbb{Z}).$ 

**then** there is $S_1 \in PSL(m, \mathbb{Z})$ and $W_1 \in GL(m,\mathbb{Z}), W_2  \in M_{m \times n}(\mathbb{Z}), W_3 \in M_{n \times m}(\mathbb{Z}) $ such that 
\begin{equation}
\left(
\begin{array}{cc}
W_1     & W_2  \\
W_3     &  W_4
\end{array}
\right)^{T}.\left(
\begin{array}{cc}
S_1     & 0 \\
0     &  S_2
\end{array}
\right).\left(
\begin{array}{cc}
W_1     & W_2  \\
W_3     &  W_4
\end{array}
\right)=\left(
\begin{array}{cc}
S_1     & 0 \\
0     &  S_2.
\end{array}
\right)
\end{equation}
I also use $W=\left(
\begin{array}{cc}
W_1     & W_2  \\
W_3     &  W_4
\end{array}
\right).$ below to shorten the notations 

**One nontrivial example**
Maybe it is a little abstract, let us see an example, given $S_2=\left(
\begin{array}{cc}
0     & \frac{1}{8} \\
\frac{1}{8}  &  0
\end{array}
\right)$ and $W_4=\left(
\begin{array}{cc}
3     & 0 \\
0  &  3
\end{array}
\right).$ One can choose the following $W$ and $S_1=\left(
\begin{array}{cc}
0     & 1 \\
1  &  0
\end{array}
\right)$
\begin{equation}
    W=\left(
\begin{array}{cccc}
 3 & 0 & 0 & 8 \\
 0 & 3 & -8 & 0 \\
 0 & -1 & 3 & 0 \\
 1 & 0 & 0 & 3 \\
\end{array}
\right)
\end{equation}
One can check the following is True.
\begin{equation}
    W^{T}.\left(
\begin{array}{cccc}
 0 &1 & 0 & 0 \\
 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 8 \\
  0 & 0 & 8 & 0 \\
\end{array}
\right).W=\left(
\begin{array}{cccc}
 0 &1 & 0 & 0 \\
 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 8 \\
  0 & 0 & 8 & 0 \\
\end{array}
\right)
\end{equation}

My conjecture is that always works for any $W_4$ and $ S_2.$ In other words, even though sometimes, we can not have $W_4^{T}S_2W_4=S_2,$ but we can provide extra space for our $W_4$ to embed it into larger integer matrix $W \in PSL(n+m.\mathbb{Z})$ so that if we focus on the subspace part $S_2$. We still get some kinds of preservation. I also require $det(S_1)=1.$ One can ask the same question by adding constraints that $W_4$ and $S_2$ are symmetric. But I believe the conjectures are always true if we allow $m$ to be arbitrarily large.
**I have searched the literature. However, except for using computers, I do not have any systematic way to tackle this conjecture.**

In terms of integer lattice language, we can define an integer lattice $S_2^{-1}$ and I want to claim that there is another unimodular $det(S_1)=1.$ lattice $S_1$ such that one can direct sum two lattice $S_1+S_2$. Then that combined lattice has an automorphism group such that its corner (or focus on $S_2$ part) can realize that $W_4$.

**Special case**
if $W_4 \in Aut(S_2)=\{x|x \in GL(n, \mathbb{Z}), x^{T}S_2x=S_2. \}$ [see related][1]. Then the conjecture is true by choosing $S_1$ and all $W_1,W_2. W_3$ to be empty. The original equation already holds.
This is the simplest case. One can further require $S_2$ to be positive definite.

**One Suspicion**
I suspect that when the difference of $W_4^{T}S_2W_4-S_2=x \in GL(n, \mathbb{Z}).$ The conjecture should be easily proved.

**Simplest nontrivial case:**
However, for the following example, $S_2=\frac{1}{8}, W_4=3.$ I can not find any solutions. Simplest nontrivial case: Even I required $W_4, S_2$ are just integers. In this case, $S_2=\frac{1}{p}, p \geq 2.$ ansd suppose $W_4^2=1(mod p).$ I can not prove it.

Any suggestions or ideas are really welcome. Feel free to stronger the condition and prove any case except for the trivial one are really welcome.


  [1]: https://mathoverflow.net/questions/378780/variation-of-centraliser-in-operatornamegln-mathbbz