Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations

If L = \sum_{j=1}X_j^2 + X_0 is a Hormander partial differential operator in R^N, we give sufficient conditions on the X_j's for the existence of a Lie group structure G = (R^N,*), not necessarily nilpotent, such that L is left invariant on G. We also investigate the existence of a global fundamental solution Gamma for L, providing results ensuring a suitable left invariance property of. Examples are given for operators L to which our results apply: some are new, some appear in recent literature, usually quoted as Kolmogorov-Fokker-Planck type operators.