After defining a commutator function
Commutator[X_, Y_] := X.Y - Y.X;
I would like to be able to simplify abstract matrix algebra expressions such as
Simplify[Commutator[A + B, Commutator[A + B, A - B]] -
Commutator[A - B, Commutator[A + B, A - B]]]
instead of just printing out
-(A - B).(-(A - B).(A + B) + (A + B).(A - B)) + (A +
B).(-(A - B).(A + B) + (A + B).(A - B)) + (-(A - B).(A + B) + (A +
B).(A - B)).(A -
B) - (-(A - B).(A + B) + (A + B).(A - B)).(A + B)
Is there a way to do this without downloading a non-commutative algebra package?