Let $G$ be a Lie group and $M$ be a manifold such that $G$ acts on $M$ transitively. Let $$\phi: G\times M\to M$$ be the action map. Now $\phi $ induces a homomorphism $$\alpha:G\to Aut(M)$$
How to show that $\alpha$ induces a Lie homomorphism $\gamma:\mathfrak g\to \Gamma(M,TM )$? Is $\gamma$ injective? what if the action is effective?