Superspace Yang-Mills Bianchi identity ($D^{\alpha }{W}_{\alpha }={\bar{D}}_{\dot{\alpha }}{\bar{W}}^{\dot{\alpha }}$)

For a real Abelian vector superfield, $W_{\alpha }=-{\bar{D}}^2{D}_{\alpha }V/4$ is chiral. The supercovariant derivative algebra with left derivatives gives $D^{\alpha }{\bar{D}}^2{D}_{\alpha }={\bar{D}}_{\dot{\alpha }}{D}^2{\bar{D}}^{\dot{\alpha }}$, proving the displayed reality condition. Its component equations contain the ordinary gauge-theory Bianchi identity, not the Maxwell equations of motion. For a non-Abelian gauge group, replace these by gauge-covariant derivatives and compare both field strengths in the same gauge frame; the identity follows from the graded Jacobi identity of the superconnection.