I am interested in infinite order elements $A\in SL(3, {\mathbb Z})$ whose spectra are not contained in ${\mathbb R}$ (i.e. such $A$ has two distinct complex-conjugate eigenvalues which are not roots of unity); I will refer to them as NRS matrices.

Question. Is there a pair of commuting NRS matrices $A, B\in SL(3, {\mathbb Z})$ whose product is again an NRS matrix, such that $A, B$ generate a non-cyclic subgroup of $SL(3, {\mathbb Z})$?

As the last resort, one can look for such matrices by computer-search, but I would prefer to avoid doing this.