The concrete problem is this: Find triplets of distinct matrices $(A,B,C)$ of dimension $6\times 6$ over the field $\mathbb{F}_{2^2}$ such that:
- $A^2B=AB^2$
- $C^2A=CA^2$
- $B^3C=BC^3$
However, I'm also interested in how this can be done in a more general setting at least with other sets of equations; for now I can assume I always need triplets of square matrices of small dimension over a small finite field.
If it can be done with a well-known computer algebra system (esp. SAGE) it will be great.