It is shown that in the four-dimensional, source-free Einstein–Maxwell theory, four distinct orthogonal eigenvectors of the energy-momentum tensor may be obtained formally by the action of four projection operators on an almost arbitrary vector.

The extremal field, of the unified theory of Maxwell, Einstein and Rainich, is obtained explicitly in terms of eigenvectors of the energy-momentum tensor and the energy-momentum tensor itself, expressed in terms of eigenvectors, is seen to take on two equivalent forms. From this, a two-parameter group of Lorentz rotations, which leave the tetrad of eigenvectors unchanged, is deduced.