= Power of the Yang-Mills invariant
{title2=$\mathcal L=s^p,\quad s=\kappa(F_{\mu\nu},F^{\mu\nu})$}
For a semisimple gauge algebra and positive integer $p$, this <gauge-invariant> density yields $D_\mu(s^{p-1}F^{\mu\nu})=0$. The curvature variation $\delta F_{\mu\nu}=D_\mu\delta A_\nu-D_\nu\delta A_\mu$ gives $\delta s=4\kappa(D_\mu\delta A_\nu,F^{\mu\nu})$, and <gauge-covariant integration by parts> proves the equation. More generally a density $f(s)$ gives $D_\mu(f'(s)F^{\mu\nu})=0$. The product form retains the equation at zeros of $s$; dividing by $s$ can discard solutions.
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