  In this paper we discuss the consequences of a Killing symmetry on the local
geometrical structure of four-dimensional spacetimes. We have adopted the point
of view introduced in recent works where the exterior derivative of the Killing
plays a fundamental role. Then, we study some issues related with this approach
and clarify why in many circumstances its use has advantages with respect to
other approaches. We also extend the formalism developed in the case of vacuum
spacetimes to the general case of an arbitrary energy-momentum content.
Finally, we illustrate our framework with the case of spacetimes with a
gravitating electromagnetic field.
