What is the main use of Lie brackets in the Lie algebra of a Lie group? 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!

A good question. There are many aspects of the situation... At least one fundamental structure can be understood in the following way. First, imagining that $t$ is an "infinitesimal", so that $t^3=0$ (not $t^2=0$!) (or equivalent...), and imagining that elements of the Lie group near the identity are $g=1+tx$ and $h=1+ty$ (with $x,y$ in the Lie algebra) observe that $(1+tx)(1+ty)(1-tx)(1-ty)=1+t^2(xy-yx)$. Thus, we care about $xy-yx=[x,y]$.
E.g., for matrix Lie groups, so that $x,y$ are matrices, this makes sense, where $xy-yx$ is in the matrix algebra.
Yes, several issues are left hanging after this walk-through, but the symbol-pattern proves to be excellent, in essentially all incarnations.
