Supercovariant derivative algebra with left derivatives ($\{D_{\alpha },{\bar{D}}_{\dot{\beta }}\}=2i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\partial }_{\mu }$)

For left Grassmann derivatives, choose $D_{\alpha }={\partial }_{\alpha }+i{\sigma }_{\alpha \dot{\beta }}^{\mu }{\bar{\theta }}^{\dot{\beta }}{\partial }_{\mu }$ and ${\bar{D}}_{\dot{\alpha }}={\bar{\partial }}_{\dot{\alpha }}+i{\theta }^{\beta }{\sigma }_{\beta \dot{\alpha }}^{\mu }{\partial }_{\mu }$. Differentiating the other operator's odd coefficient gives two equal contributions to the mixed anticommutator; coefficient-coefficient terms cancel by the Grassmann algebra. Thus the mixed anticommutator is $2i{\sigma }^{\mu }{\partial }_{\mu }$, while equal-chirality ones vanish. Also $\bar{D}(x+i\theta \sigma \bar{\theta })=0$, giving the coordinates used in the chiral-superfield component expansion. Negating the barred derivative negates the mixed algebra.