Zassenhaus neighbourhood

Remarks

Every Lie group has a Zassenhaus neighbourhood, by a theorem of Zassenhaus. One can moreover choose $U$ so that $\Gamma \cap U$ lies in a connected nilpotent Lie subgroup of $G$.

Example: in $G = \R^n$, every open neighbourhood of $0$ is a Zassenhaus neighbourhood. Indeed every subgroup of $\R^n$ is abelian, hence nilpotent.

Non-example: $U = \mathrm{SO}(3)$ in $G = \mathrm{SO}(3)$. The rotation group of the icosahedron is a finite, hence discrete, subgroup isomorphic to the alternating group $A_5$. It lies in $U$ and is not nilpotent, since $A_5$ is simple and non-abelian.