Completing Lie algebra actions to Lie group actions

For a finite-dimensional Lie algebra $\mathfrak {g}$ of vector fields on a manifold $M$ we show that $M$ can be completed to a $G$-space in a universal way, which however is neither Hausdorff nor $T_1$ in general. Here $G$ is a connected Lie group with Lie-algebra $\mathfrak {g}$. For a transitive $\mathfrak {g}$-action the completion is of the form $G/H$ for a Lie subgroup $H$ which need not be closed. In general the completion can be constructed by completing each $\mathfrak {g}$-orbit.