A global lifting for finite-dimensional lie algebras of complete vector fields

Motivated by the so-called Lifting and Approximation Theorem by Rothschild and Stein, we consider a set of vector fields [Formula: see text] on a manifold [Formula: see text], and we study the problem of obtaining a global lifting of the [Formula: see text]’s to a system of generators of the Lie algebra [Formula: see text] of a Lie group [Formula: see text]. By assuming that the Lie algebra [Formula: see text] generated by [Formula: see text] is finite-dimensional and all of the [Formula: see text]’s are complete vector fields, but without the assumption that they satisfy Hörmander’s rank condition, we reduce the lifting problem to a result of Palais on integrability. This proves that any [Formula: see text] in [Formula: see text] is [Formula: see text]-related to a left invariant vector field [Formula: see text], where [Formula: see text] is a smooth map resulting from a right action [Formula: see text] of [Formula: see text] on [Formula: see text]. Both the lifting map [Formula: see text] and the lifting vector fields [Formula: see text] are globally defined, and our result generalizes the global Lifting obtained by Folland in the special case of dilation-invariant vector fields. According to Palais’ Integrability, the map [Formula: see text] is obtained via the flow of suitable vector fields on [Formula: see text] and the image set of [Formula: see text], namely, [Formula: see text], is the Sussmann orbit of [Formula: see text] through [Formula: see text]. Examples are provided, showing that a germ of [Formula: see text], obtained through the integration of [Formula: see text] and depending only on the Baker–Campbell–Hausdorff formula, is often sufficient to get the global lifting without the need of abstract results.