  In their celebrated paper of 1976, Rothschild and Stein prove a lifting
procedure that locally reduces to a free Lie algebra any family of smooth
vector fields $X_1,\dots,X_q$, over a manifold $M$ respecting the H\"ormander's
condition. Then, a large class of differential operators can be lifted, and
they find that fundamental solutions on the lifted space can be re-projected to
fundamental solutions of the given operator on $M$. In case that the Lie
algebra $\mathfrak g=\mbox{Lie}(X_1,\dots,X_q)$ is finite dimensional but not
nilpotent, this procedure could introduce a strong tilting of the space. In
this paper we represent a global construction of a Lie group $G$ associated to
$\mathfrak g$ that avoid this tilting problem. In particular
$\mbox{Lie}(G)\cong\mathfrak g$ and a right $G$-action exists over $M$,
faithful and transitive, inducing a natural projection $E\colon G\to M$. We
represent the group $G$ as a direct product $M\times G^z$ where the model fiber
$G^z$ has a group structure. We prove that for any simply connected manifold
$M$ -- and a vast class of non-simply connected manifolds -- the existence of a
fundamental solution for a differential operator $L=\sum_{\alpha\in\mathbb N^q}
r_\alpha\cdot X^\alpha$ of finite degree over $M$, follows via a saturation
method from the existence of a fundamental solution for the associated lifted
operator over a group $G$. This is a generalization of Biagi and Bonfiglioli
analogous result for homogeneous vector fields over $M=\mathbb R^n$.
