  In this paper we discuss function spaces on a general noncompact Lie group,
namely the scales of Triebel--Lizorkin and Besov spaces, defined in terms of a
sub-Laplacian with drift. The sub-Laplacian is written as negative the sum of
squares of a collection of left-invariant vector fields satisfying
H\"ormander's condition. These spaces were recently introduced by the authors.
In this paper we prove a norm characterization in terms of finite differences,
the density of test functions, and related isomorphism properties.
