Before revision: For mathematical details see Spin group The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's valence electron shell and to complexes of ions like Cu2+ which have a single "vacancy" in the valence shell.
After revision: For mathematical details see Direct product of groups The concept of a double group was introduced by Hans Bethe for the quantitative treatment of magnetochemistry of complexes of ions like Ti3+, that have a single unpaired electron in the metal ion's valence electron shell and to complexes of ions like Cu2+ which have a single "vacancy" in the valence shell.