Can someone evaluate whether the following is true:
Given two symmetric, commuting matrices $M_1,M_2 \in \mathrm{SL}(3,\mathbb{Z})$ (integer entries with determinant 1), where $M_2$ is not expressible as $M_1$ raised to an integer power. If $M_1$ and $M_2$ commute, any symmetric matrix $M_3 \in \mathrm{SL}(3,\mathbb{Z})$ that commutes with $M_1$ and $M_2$ be written as $M_3=M_1^iM_2^j$ for some integers $i$ and $i$.
My suspicion is that the answer is yes, but I don't think I have the right tools to say so conclusively.
No. We can take $$ M_1=1, \quad M_2=\begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{pmatrix} , \quad M_3=\begin{pmatrix} 1 & 0 \\ 0 & A \end{pmatrix} , $$ with $A\in SL(2,\mathbb Z)$ symmetric, but not a multiple of the identity.