**Abstract** : We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We therefore reconsider the interpretation to see if we can render it consistent also with the symmetry. The results confirm clearly that the traditional physical interpretation of the anomalous Zeeman effect is not correct. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. This is even true for charged particles " without spin ". Particles " with spin " acquire additional spin in a magnetic field. This additional spin must be combined algebraically with the pre-existing spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but can be understood in terms of the Berry phase on a path through the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to " shut up and calculate ".