Proposition: A necessary quotient condition

An odd derivation $D$ of an off-shell algebra induces a derivation on its quotient by an ideal $\mathcal{I}$ only if $D\mathcal{I}\subseteq \mathcal{I}$. The equation $D^2=0$, even off shell, does not imply this condition.

Proof. If $F$ and $F+J$ represent the same quotient element, their proposed images differ by $DJ$. Well-definedness requires $DJ\in \mathcal{I}$. Nilpotence tests a different composition and imposes no such requirement.