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How can one prove the Bianchi identity of a non-Abelian gauge theory? i.e. $$ \epsilon_{\mu \nu \lambda \sigma}(D_{\nu}F_{\lambda \sigma})^a=0 $$

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2 Answers 2

Hint: Use the Jacobi identity $$\sum_{\mu,\nu,\lambda~{\rm cycl.}} [D_{\mu},[ D_{\nu}, D_{\lambda}]]~=~0. $$

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Hint: use $$ [D_{\nu},D_{\mu}] = F_{\nu \mu} $$

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