Left-acting BRST differential ($s(UV)=(sU)V+(-1{)}^{|U|}U(sV),s^2=0$)

Writing a transformation as $\delta \Phi =ϵs\Phi$ with the odd parameter on the left defines this graded derivation convention. For Yang-Mills fields, $sA=Dc$, $sc=-[c,c]/2$, $sb=h$, and $sh=0$. The Jacobi identity supplies nilpotence; the placement of the odd parameter controls signs in a Noether current.