One real function instead of the Dirac spinor function

Three out of four complex components of the Dirac spinor can be algebraically eliminated from the Dirac equation (if some linear combination of electromagnetic fields does not vanish), yielding a partial differential equation of the fourth order for the remaining complex component. This equation is generally equivalent to the Dirac equation. Furthermore, following Schr\"{o}dinger (Nature, \textbf{169}, 538 (1952)), the remaining component can be made real by a gauge transform, thus extending to the Dirac field the Schr\"{o}dinger's conclusion that charged fields do not necessarily require complex representation. One of the two resulting real equations for the real function describes current conservation and can be obtained from the Maxwell equations in spinor electrodynamics (the Dirac-Maxwell electrodynamics). As the Dirac equation is one of the most fundamental, these results both belong in textbooks and can be used for development of new efficient methods and algorithms of quantum chemistry.