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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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    Thanks Sasha, for your comment! I am interested in the third Lie theorem using Levi-Malcev decomposition of a Lie algebra $\mathcal{G}$ as a semi-direct sum of a semi-simple sub algebra S and its radical $\mathfrak{rad}$ (maximal solvable ideal). In particular if $\mathcal{G}=SL(2)\ltimes H_3$ where $H_3$ is isomorphic to the Heisenberg Lie algebra, then what is the connected and simply connected Lie group associated to $\mathcal{G}$?2011-08-11

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