 The same classification is used for SL(2,C) and PSL(2,C) (Möbius transformations) and PSL(2,R) (real Möbius transformations), with the addition of "loxodromic" transformations corresponding to complex traces; analogous classifications are used elsewhere.A subgroup that is contained with the elliptic (respectively, parabolic, hyperbolic) elements, plus the identity and negative identity, is called an elliptic subgroup (respectively, parabolic subgroup, hyperbolic subgroup).This is a classification into "subsets," not "subgroups