On-shell closure of rigid worldsheet supersymmetry ($[{\delta }_1,{\delta }_2]=2{\overline{ϵ}}_1{\rho }^{\alpha }{ϵ}_2{\partial }_{\alpha }$)

For $\delta X=-\overline{ϵ}\psi$ and $\delta \psi ={\rho }^{\alpha }{\partial }_{\alpha }Xϵ$, with constant Grassmann-odd Majorana parameters, the commutator on $X$ is the displayed translation. Set $M={ϵ}_1{\overline{ϵ}}_2-{ϵ}_2{\overline{ϵ}}_1$. In two dimensions $\{{\rho }^{\alpha },M\}=2{\overline{ϵ}}_1{\rho }^{\alpha }{ϵ}_2$ times the identity, so the fermion commutator is the translation minus $M{\rho }^{\alpha }{\partial }_{\alpha }\psi$. The remainder vanishes on the fermion field equation. This distinguishes on-shell closure from the off-shell total-derivative invariance of the action.