Every Lie group $G$ has the following escape property: For every $x \ne e$ in a sufficiently small neighborhood $U$ of the identity $e$ in $G$, there is a integer $n$ such that $x^n$ is not in $U$.

The $\textbf{Question one }:$ is if we can find a sufficiently small neighborhood $V$ of $e$ in $G$, for any two different points $a,b \in V$ , there is a integer $m$ such that $a^m(b^{-1})^m$ is not in $V$.

Thanks for Scholar's answer about the above question. Again, I continue to consider a more eccentrica behavior.

$\textbf{Question Two}:$ For compact lie group $G$, Can we find a sufficiently small neighborhood $V$ of $e$ in $G$, for any two different points $a,b \in V$ , there exist $n$ integers, denote $k_1,k_2,k_3,,,,k_n$ such that if n is odd $a^{k_1}b^{k_2}a^{k_3}....a^{k_n}(b^{k_1}a^{k_2}b^{k_3}....b^{k_n})^{-1} \in V$, if n is even, $a^{k_1}b^{k_2}a^{k_3}....b^{k_n}(b^{k_1}a^{k_2}b^{k_3}....a^{k_n})^{-1} \in V$?

Because I don't know the properties of groups.Maybe this problem is very complicated. You can look at it casually. Thank to for all Scholars to answer my question. Your suggestions help me understand some own mistake.Thanks.

If $G$ is a commutative group the question actually has a positive answer, since $G$ has the escape property. I guess it is true for some larger class of Lie groups. The problem is to state that the power mapping can enlarge the distance between two different points. But I didn't find a good property of power mapping by looking up data.

meanwhile，Thanks Michael Albanese for editing of the question.