SL _{2, C} symmetry of the gravitational field

In a previous paper the symmetry of the gravitational-field dynamical variables was discussed. A scheme was given according to which the spin coefficients and the Riemann tensor were represented in the form of linear combinations of the infinitesimal generators of the group SL _{2, C}, similar to the way Yang and Mills write their dynamical variables in terms of the Pauli spin matrices, where the spin coefficients take the role of the Yang-Mills-like potentials and the Riemann tensor takes the role of the fields. In this paper we use this representation of the gravitational-field dynamical variables in order to write the field equations of general relativity, and obtain a simple and attractive representation for the field equations of Newman and Penrose. Comparison between the present method and that of Utiyama, based on the Yang-Mills theory, is discussed and the possibility of using it as a basis for quantization is briefly mentioned.