  We show that the Dirac equation for real spinors can be naturally decomposed
into a system of two first-order relativistic wave equations. The decomposition
separates in a transparent way the real and imaginary parts of the Dirac
equation by means of two algebraic differential operators, allowing to describe
real spinors in any representation of the Dirac matrices maintaining the
reality condition $\tilde{\Psi}=\tilde{\Psi}^{*}$ unaltered. In addition, it is
shown that the Majorana wave equation is a particular case of the relativistic
system of equations deduced in this paper. We also briefly discuss how the
formalism can be extended to deal with complex (charged) spinors.
