Abelian field-strength chiral projection ($W_{\alpha }=-\frac{1}{4}{\bar{D}}^2{D}_{\alpha }V$)

For an Abelian vector superfield, the chiral field-strength superfield is $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$. Three barred supersymmetric covariant derivatives vanish, proving ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. This is a chiral spinor superfield with a gaugino as its lowest component. Phase and component signs depend on the stated Wess-Zumino gauge convention; the reality of $V$ alone does not fix those phase conventions.