I would like to derive the Einstein-Skyrme equations. The action can be read as

\begin{equation} S[g,U] = \int d^{4}x\sqrt{-g}\biggl[R + \frac{K}{4}Tr\bigg(A^{\mu}A_{\mu} + \frac{\lambda}{8}F_{\mu\nu}F^{\mu\nu}\bigg)\biggr] \end{equation} where the constants $K$ and $λ$ are the ones corresponding to the Skyrme coupling, and $A_{\mu}=U^{-1}\nabla_{\mu}U$ with $U\in SU(2)$ and strength of the field is $F_{\mu\nu}=[A_{\mu},A_{\nu}]$. The equation that I do not know how to get is associated to the variation of $A_{\mu}$

\begin{equation} \nabla^{\mu}A_{\mu} + \frac{\lambda}{4}\nabla^{\mu}[A^{\nu},F_{\mu\nu}]=0. \end{equation}


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