  In this paper we establish a number of geometrical inequalities such as
Hardy, Sobolev, Rellich, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg,
Gagliardo-Nirenberg inequalities and their critical versions for an ample class
of sub-elliptic differential operators on general connected Lie groups, which
include both unimodular and non-unimodular cases in compact and noncompact
settings. We also obtain the corresponding uncertainty type principles.
