  The goal of this paper is to express the Bach tensor of a four dimensional
conformal geometry of an arbitrary signature by the Cartan normal conformal
(CNC) connection. We show that the Bach tensor can be identified with the
Yang-Mills current of the connection. It follows from that result that a
conformal geometry whose CNC connection is reducible in an appropriate way has
a degenerate Bach tensor. As an example we study the case of a CNC connection
which admits a twisting covariantly constant twistor field. This class of
conformal geometries of this property is known as given by the Fefferman metric
tensors. We use our result to calculate the Bach tensor of an arbitrary
Fefferman metric and show it is proportional to the tensorial square of the
four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills
equations imposed on the CNC connection for all the homogeneous Fefferman
metrics. The only solution is the Nurowski-Plebanski metric.
