Lemma: Kinematical nilpotence

The rule (eq:skin) is nilpotent on the off-shell graded tensor differential algebra. This is not a claim that it acts on an interacting quantum algebra.

Proof. For $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$, expansion with the odd Leibniz rule gives ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. The terms with symmetric second spacetime derivatives cancel against the antisymmetric ghost coefficients. Thus

The scalar calculation is the same. In ${s_{\mathrm{kin}}}^{2}c$, the two terms involving first derivatives cancel after relabelling, and the second-derivative term is zero by ghost antisymmetry. The nonminimal pair is nilpotent by definition.