Suppose $G$ is a connected, simply-connected, complex Lie group and $\mathfrak{g}$ is the Lie algebra of $G$. Suppose $X$ is a complex manifold (we can assume it is compact and connected, if needed). I want to show if $\rho:\mathfrak{g} \to H^{0}(X,TX)$ is a Lie algebra homomorphism, then there exists a (right)-group action $\sigma: G \times X \to X$ such that associated Lie algebra homomorphism to $\sigma$ is $\rho$, where the associated Lie algebra homomorphism is defined in
Group action induces a homomorphism between Lie algebras
This is used in a paper I am reading. The problem I am having is that, I don't know how use Lie groups and Lie algebras correspondence
https://en.wikipedia.org/wiki/Lie_group%E2%80%93Lie_algebra_correspondence#The_correspondence
to get the result I want.
This is called the Lie-Palais Theorem, and was originally proved locally by Lie, and in global form by Palais in 1957. Apart from the original reference, another place you can find an exposition is "Structure and Geometry of Lie Groups" by Neeb and Hilgert, Section 10.5.