Find the matrix of $2\times 2$ that commute with $(AB-BA)^2$.

Question

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 $A,B,C\in R$.

Answer 1

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.