Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie algebra associated to an Abelian group. We have the following relation between these generators $$[Q^a,P^b]=if^c_{ab}P^c$$
I would like to know what we can say above the following trace. Is it equal to zero? $$tr([Q^a,P^b]Q^c P^d)$$
Cheers!