no. it's easier to work with matrices rather than transformations here. they're the same anyway: (here has 1 as its (i,j)-entry and 0 everywhere else.)
If A,B,C are linear transformations on a finite dimensional vector space, then does (AB - BA)^2 commutes with C always?
in this case, the claim is true: let clearly thus by Cayley Hamilton: for some scalar now it's obvious that for all
What happens when the dimension of the vector space is 2?