I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at the identity, so that $\mathfrak g$ have a very interesting property : linearity.
However $\mathfrak g$ has another property : it is stable under Lie brackets $[.,.]$.
For me when I study Lie groups I always find linearity of Lie algebras really important, and I don't see and I didn't find why the stability under Lie brackets is important. What is the main result/property of Lie groups using this property?
That would be great if you could light me!