In class it was shown that $$ i[Q_\epsilon,T^{\mu\nu}] = -(\epsilon\cdot\partial)T^{\mu\nu} - \partial_\rho (\epsilon^\mu T^{\rho\nu}) + \partial^\nu(\epsilon_\rho T^{\rho\mu}) $$ with $$ Q_\varepsilon = \int dS_\mu \varepsilon_\nu T^{\mu\nu} $$

We are then asked to show that this implies $i[Q_{\varepsilon_1},Q_{\varepsilon_2}]=Q_{[\epsilon_1,\epsilon_2]}$. I haven't felt this stuck for a while, and hope that somebody could elucidate parts of this problem.

In a first part of the question, it was claimed that it is "easily seen" that the LHS of the uppermost equation (in this question) obeys $$ i[Q_\epsilon,T^{\mu\nu}] =-((\epsilon\cdot\partial )T^{\mu\nu}+(\partial\cdot \epsilon)T^{\mu\nu} + \partial^\nu(\epsilon_\rho T^{\rho}_\mu)) $$

I could not derive any of the equalities, and upon asking about them, I was referred to Blumenhagen, Rychkov & Qualls in the books on CFT. I could not figure it out using these sources and hope somebody could elucidate how to tackle the problem. How does the "easily seen" step lead to the equality? And what is a good first step to show the requested implication?


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