Working on Lie algebra basics, I found the following property: let $G$ be a closed subgroup of $GL_n$ over $\mathbb{R}$. Let $\mathfrak{g}$ its Lie algebra and $\mathfrak{m}$ a supplementary space. Then
There is a neighborhood $V$ of $0$ in $\mathfrak{m}$ such that $\exp(V) \cap G = \{I_n\}$.
I have the feeling that this could lead to prove the usual property stating that $GL_n$ has no arbitrarily small subgroups. Is it possible? (or is it totally unrelated?)