= Variation of the Chern-Simons 5-form
{title2=$\delta Q_5=3\operatorname{Tr}(\delta A\,F^2)+d\Theta$}
The boundary four-form is $\Theta=\operatorname{Tr}[\delta A\,(FA+AF-\tfrac12A^3)]$. To derive the identity, put $B=dA$ and $a=\delta A$. The variations of the three polynomial terms in $Q_5$ are respectively $3\operatorname{Tr}(aB^2)+d\operatorname{Tr}[a(BA+AB)]$, $3\operatorname{Tr}[a(A^2B+BA^2)]+\tfrac32d\operatorname{Tr}(aA^3)$, and $3\operatorname{Tr}(aA^4)$. Their sum gives the identity because $F=B+A^2$. For variations in a <Lie algebra> with basis $T_\alpha$, the bulk field equations are $\operatorname{Tr}(T_\alpha F^2)=0$, not necessarily the unprojected matrix equation $F^2=0$. Conjugation of both the Lie algebra and the <gauge curvature> proves covariance of these equations under <Yang-Mills gauge transformations>.
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