To decompose the left regular representation of G on L 2 ( G ) only the principal series are required. This immediately yields the decomposition on the subrepresentations L 2 ( G / { ± ± I } ) , the left regular representation of the Lorentz group, and L 2 ( G / K ) , the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with k = 0.)