Bethe's early group-theoretical treatment is supplemented by some physical arguments illustrating his method of semiquantitative evaluation of the term splitting. Up to date notations are introduced. The cases of weak, intermediate, and strong crystal-field splitting are discussed, with typical ions given as examples. For cubic and hexagonal site symmetry, the Bethe method is demonstrated step-by-step and the results applied to transition metal ions. Double-valued group representations, double groups and Kramers degeneracy are introduced, compared, and illustrated with physical examples. Certain mathematical analogies between formulas of representation theory and angular momentum theory are pointed out.