Classical geometrodynamics revisited

Classical geometrodynamics, the ''already unified'' theory of electromagnetism and gravitation, is rewritten in the complex bivector formalism. The null as well as nonnull case are treated. It is found that, in both cases, a one-form constructed with the spin coefficients naturally arises and masters the theory. The vanishing of its exterior derivative must be included in the necessary and sufficient conditions for a metric to be a solution of the source-free Einstein-Maxwell equations