 Self-duality of gauge curvature

In oriented Euclidean four-space, gauge curvature is self-dual when $F=\ast F$. The Hodge star squares to one on two-forms, so this picks its positive eigenspace; the negative eigenspace is anti-self-duality of gauge curvature. The Bianchi identity then implies the Yang-Mills equations, and the action saturates the positive-charge Yang-Mills instanton Bogomolny bound.