Suppose $A,B\in SL(3,F_q)$, where $F_q$ is the finite field of order $q$ and $SL(3,F_q)$, the group of matrices with determinant one and entries from $F_q$ , are such that $A$ has eigenvalues in $F_q$ and $B$ has eigenvalues in $\overline{F_q}\setminus F_q$. Also, $A$ is diagonalizable over $F_q$ and $B$ is diagonalizable over $\overline{F_q}$, the closure of $F_q$. I am trying to show that $A$ and $B$ are not simultaneously diagonalizable by a matrix $P\in SL(3,\overline{F_q})$ (i.e., $\nexists P\in SL(3,\overline{F_q})$ such that $(PAP^{-1},PBP^{-1})$ are diagonal). I am considering the approach of looking at the $F_q$ algebra generated by $A$ and $B$ and trying to show it is not isomorphic to the algebra generated by $F_q[PAP^{-1},PBP^{-1}]$. I am looking for some reference which might be helpful in proving the above. Most of the results I saw had been about irreducible representations which is not helpful for my case. I would appreciate it if you could suggest some reference that could be helpful. Thanks in advance for your time.