This is worked out for one of the lines of the Zeeman effect in the 2p to 3d transition of atomic H at 656.3 nm. The relativistic splitting is in the megahertz range and is given by Eq. (17). The relativistic effect in linear polarization splits the central Zeeman line into three. Zeeman could not have observed this effect because his resolution was not nearly enough. There will be different patterns of relativistic splitting for every atom and molecule, and this is a solution of the Dirac equation in the presence of a magnetic field. The effect can be observed with visible / radio frequency double resonance or similar technique or by adapting an ESR spectrometer. This is a completely new Zeeman spectroscopy discovered simply by using the relativistic momentum instead of the classical momentum. It is a new and precise test of the Dirac equation or fermion equation.