Lie group action from infinitesimal action

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.

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