Calculating an expression for trace of generators of two Lie algebra

[vnikoofard]

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?
$$tr([Q^a,P^b]Q^c P^d)$$

Comment:
From the cyclic property of trace we have
$$tr[A,B]=0$$
for any matrices. Also
$$tr([A,B]C)=0$$
just for symmetric matrices. Maybe these relations help!
Cheers!

 

[fresh_42]

Some quick calculations let me assume that this is not true. You could check the Poincaré algebra.