What is the induced action on the Lie algebra of a Lie group

Question

Suppose $G$ is a Lie group and $\Gamma$ be its Lie subgroup. If $\Gamma$ acts on $G$, does it also acts on the Lie algebra of $G$? Assuming that it induces an action on the Lie algebra of $G$ and that $G$ is the matrix Lie group how one computes the induced action. I came across this question while reading this paper by Baues. Any help is appreciated. I am new to linear algebraic group so pointing to where can I learn these stuffs will also help.