This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).

Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i.e. $M$ is a unipotent matrix).

Write $M=\begin{bmatrix} A_1&A_2\\ A_3&A_4 \end{bmatrix}, $ where the $A_i$ are $2$ by $2$ sumbatrix of $M$.

Let $d_i$ be the dot product of two rows of $A_i$, i.e. if $A_i = \begin{bmatrix} a&b\\ c&d \end{bmatrix}$, then $d_i = ac +bd$.

Let $a_i = \mathrm{det}(A_i) - d_i$.

For example if $A_1 =\begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}$, then $d_1 = 11$ and $a_1 = - 2 - 11 = -13$.

Consider the matrix $A = \begin{bmatrix} a_1&a_2\\ a_3&a_4 \end{bmatrix}$.

**Question**: Suppose $A$ is in $\mathrm{GL}(2, \mathbb{Z})$, can one of eigenvalues of $A$ have absolute value bigger than $1$?

**Thoughts so far**: It is clear that the matrix $A$ should have all entries being non-zero. I find this question hard because two similar matrices might have differnt "behaviour", e.g. if $
M_1=\Tiny\begin{pmatrix} 0 & 0 & -1 & 0 \cr
1 & 0 & 0 & 1 \cr 1 & 0 & 0 & 2 \cr 0 & 1 & -3 & 4 \end{pmatrix}
$, then $
A=\begin{pmatrix} 0 & -1 \cr 1 & 2 \end{pmatrix}\in \mathrm{SL}_2(\Bbb Z);
$
while if $
M=\Tiny\begin{pmatrix} 1 & 1 & 0 & 0 \cr
0 & 1 & 1 & 0 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 0 & 1 \end{pmatrix}
$, we have $A=\begin{pmatrix} 0 & 0 \cr 0 & 0 \end{pmatrix}$, even though $M_1$ is simliar to $M_2$.

I also look up the Block matrix determinant, but I don't think we can use the identities since our block matrices don't satisfy their assumptions.

Thank you for reading, any reference for this would be really appreciated.