Unitary Representations of the Group O(2, 1) in an O(1, 1) Basis

We consider the unitary irreducible representations of the group SO(2, 1), belonging to the continuous and the discrete classes. We cast them into a form in which the noncompact generator of an O(1, 1) subgroup is diagonal. We examine some properties of the remaining generators in this basis. We recover the known result that the spectrum of the noncompact generator covers the real line twice for representations of the continuous class, and once for those of the discrete class.