Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D, D]]?D and [K, [K, K]]?K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve.