= Derivative-squared Yang-Mills action
{title2=$\mathcal L=\kappa(D_\mu F_{\nu\rho},D^\mu F^{\nu\rho})$}
This <gauge-invariant> higher-derivative density has equation
$$
2D_\mu D_\lambda D^\lambda F^{\mu\nu}+[F_{\rho\sigma},D^\nu F^{\rho\sigma}]=0.
$$
The extra commutator comes from varying the connection inside the <adjoint covariant derivative>: $\delta(D_\mu F_{\nu\rho})=D_\mu\delta F_{\nu\rho}+[\delta A_\mu,F_{\nu\rho}]$. Two uses of <gauge-covariant integration by parts> give the first term, and Killing-form invariance gives the second. In an abelian limit it reduces to $\Box\partial_\mu F^{\mu\nu}=0$. Keeping the covariant derivative order is essential because their commutator is curvature.
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