Find the matrix of $2\times 2$ that commute with $(AB-BA)^2$. Let $k$ be field and $R$ the ring of matrix of size $2\times 2$ with entries in $k$. Show that $(AB-BA)^2 C=C(AB-BA)^2$ for all $A,B,C\in R$. 

The $2\times 2$ matrix $AB-BA$ is of trace $0$, so of the form $$ \pmatrix{a&b\\c&-a}.$$
Then its square is
$$\pmatrix{a^2+bc&0\\0&a^2+bc}$$
which is a scalar multiple of the identity, so central.
