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$$[Q^a,Q^b]=if^c_{ab}Q^c$$

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!

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# Calculating an expression for trace of generators of two Lie algebra

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