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I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdf

Even the easy abelian one (third row)

which is

$$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$

doesn't match my calculation

Since

$$D_\rho B_{\mu\nu}=(\partial_\rho+i g B_\rho)(\partial_\mu B_\nu-\partial_\nu B_\mu)$$

the equation seems to imply that

$$B_\rho (\partial_\mu B_\nu-\partial_\nu B_\mu)=0$$

What am I missing?

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# I Covariant derivative of field strength tensor

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