I would like to ask, how to deduce a Lie group action, from infinitesimal action of its Lie algebra (the so called Lie-Palais theorem). More precisely, given a differential manifold $M$ and a Lie group $G$ with Lie algebra $\mathcal{G}$. Suppose we have a Lie algebra homomorphism $\rho : \mathcal{G}\rightarrow\mathfrak{X}(M)$ (where $\mathfrak{X}(M)$ denotes the space of vector fields of $M$).
How to deduce, from $\rho$, a smooth action $G\times M\rightarrow M\ ?$ In particular, is there a nice and elementary proof of the Lie-Palais theorem?
Thanks for you help.