The main theorem for (anti)self-dual conformal torsion-free connection

In this paper we obtain results that occur on a four-manifold of conformal torsion-free connection with all possible signatures of angular metric. It is proved that three of the four terms of the formula for the decomposition of the basic tensor are equidual, one is skew-dual. Based on this result we find conditions for (anti)self-duality of external 2-forms, which are part of components of the conformal curvature matrix. With the help of the last result, the main theorem is proved: a conformal torsion-free connection on a four-manifold with the signatures of the angular metric $s=\pm 4;0$ is (anti)self-dual if and only if the Weyl tensor of the angular metric and the exterior 2-form $\Phi _{0}^{0}$ are (anti)self-dual and Einstein and Maxwell's equations are satisfied. In particular, the normal conformal Cartan connection is (anti)self-dual iff the Weyl tensor of the angular metric is the same.