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Avoiding an imaginary connection in the Dirac Equation

In a Majorana basis, the Dirac equation for a free spin one-half particle is a 4x4 real matrix differential equation. When including the effects of the electromagnetic interaction, the Dirac equation is a complex equation due to the presence of an imaginary connection in the covariant derivative, related with the phase of the spinor. In this paper we study the solutions of the Dirac equation with the null and Coulomb potentials and notice that there is a real matrix that squares to -1, relating the imaginary and real components of these solutions. We show that these solutions can be obtained from the solutions of two non-linear 4x4 real matrix differential equations with a real matrix as the connection of the covariant derivative.