By the way, is this the GTM Herstein or UTM Herstein?
EDIT: I want to explain again why to be more clear. First, if then it is finished. If not then . Notice that if if and only if its inverse (i.e. are reversed). So it is happened that then repeating the same argument from the begining but instead of but with them reversed, i.e. we arrived at (the same approach) which is a commutator of the form so it is in .