The action map induces a Lie algebra homomorphism

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?

• Take the derivative of $\alpha$ at the identity. – Amitai Yuval Jan 21 at 9:03