It is approximately true if the external field is weak enough. It breaks down in a strong magnetic field, a phenomenon known as spin-orbit decoupling. Goudsmit and Uhlenbeck imagined it this way. Suppose there is only a single electron. The electron has a magnetic moment proportional to S , and hence we get an energy term proportional to .
Life is more complicated in multi-electron atoms.
A classical model has emerged. J precesses about B , while L and S still precess about J . J is no longer constant, since it is subject to an external torque. As it happens, the energy of a stationary state depends primarily on terms other than those in formula 8 . So when , we expect to find bunches of energy levels with the same and grouped closely together, separated slightly by an energy difference depending on .