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?