Simplify expression with abstract matrices


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?