Anti-self-duality of gauge curvature ($F=-\ast F$)

Anti-self-dual gauge curvature obeys $F=-\ast F$ in oriented Euclidean four-space. In the corresponding two-spinor convention it has vanishing symmetric primed curvature spinor. A matrix-valued two-form which is both self-dual and anti-self-dual is zero, because adding the two equations gives $2F=0$.