I'm looking for examples of matrices $u$, $v$, and $Q$ satisfying the following conditions:
- $u$, $v$, and $Q$ are relatively small matrices of the same size (perhaps 2 by 2, 3 by 3, or 4 by 4)
- $u$ and $v$ do not commute
- $vu = Quv$
- $Q$ commutes with $u$ and with $v$
Examples with specific constants would be great, but even better would be examples with the matrices containing some variables, and the best would be matrices where $u$, $v$, and $Q$ do not satisfy any polynomial relations (with constant coefficients) other than the above.