# How to derive the Einstein-Skyrme equations?

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

$$$$S[g,U] = \int d^{4}x\sqrt{-g}\biggl[R + \frac{K}{4}Tr\bigg(A^{\mu}A_{\mu} + \frac{\lambda}{8}F_{\mu\nu}F^{\mu\nu}\bigg)\biggr]$$$$ where the constants $$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}$$

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