Suppose VVV is finite-dimensional with dimV≥2\dim V \ge 2dimV≥2. Prove that there exists S,T∈L(V,V)S,T \in \mathcal L(V,V)S,T∈L(V,V) such that ST≠TSST \ne TSST=TS.
Skipping this for now since it's isomorphic to a boring computation with 2x2 matrices.