<p>This work focuses on developing an efficient numerical method to solve the relativistic hydrogen-like atom in a finite magnetic field. To this end, we derive and implement an algorithm based on Gaussian-type orbitals that exploits fermionic symmetry to accelerate the calculations and to distinguish states of different character. This is then used to investigate a novel type of mixing regime between internal and external magnetic interactions. Finally, we assess the implications for the core region of heavy elements, where spin-orbit coupling is much stronger than the effects of external magnetic fields, and the consequences for quantum chemical calculations on heavy atoms in the vicinity of white dwarfs.</p>