Show that if $A,B$ are $2\times2$ matrices, then $(AB-BA)^2$ commutes with all $2\times2$ matrices. I tried to write it all out, but it becomes really messy... Is there a more elegant way to do it? 
Note that I don't know about dimensions, vector spaces & bases yet

$AB-BA$ has trace zero, so has the form $\pmatrix{u&v\\w&-u}$.
What happens when you square such a matrix?
