Let $\varphi:G\times M\rightarrow M$ be a Lie group $G$ (effective) action on a smooth manifold $M$.
Fix $g\in G$ and let $\varphi_g(x):=\varphi(g,x)$. Then $\varphi $ induces a smooth homomorphism $\lambda: G\rightarrow Aut(M);g\mapsto \varphi_g$.
Now, if $\frak g$ is the Lie algebra of $G$ and $\Gamma(M,TM)$ the algebra of smooth vector-fields on $M$. Then, is it true that $\lambda $ induces a homomorphism $\alpha:\mathfrak g \rightarrow \Gamma(M,TM)$? How $\alpha $ is defined? Is $\alpha$ injective? Can we say that the Lie algebra of $G$ is embedded in the Lie algebra of smooth vector-fields of $M$?