$\DeclareMathOperator\SO{SO}$Recall that the *closed-subgroup theorem* ([Wikipedia link](https://en.wikipedia.org/wiki/Closed-subgroup_theorem)) says that a closed subgroup of a Lie group is a Lie group.


I am pretty sure that this theorem should have a "local" generalisation. I'll formulate it for $\SO(n)$.


**Generalisation?** Fix small $\varepsilon>0$. Let $X\subset \SO(n)$ be a closed subset contained in the closed ball $\overline B\_{e}(\varepsilon)$ centred at the identity $e$ of $\SO(n)$. Suppose $X$ has the following properties.


1. $X=X^{-1}$.
2. $X\cdot X\cdot X\cdot X\cap \overline B\_{e}(\varepsilon)=X$.


Then there exists a Lie group an $G$, a morphism $\varphi: G\to \SO(n)$ and a finite subset $S\subset \SO(n)$ such that $$X=\overline B\_{e}(\varepsilon)\cap S\cdot \varphi(\overline B\_{G,e}(10 \varepsilon)),$$


where $\overline B\_{G,e}(10 \varepsilon)$ is a small ball in $G$. In particular, $X$ is a submanifold of the ball (with finite number of connected components).


Are you aware of some statement of this kind? (the power $4$ is random, maybe it should be replaced by some other power)



A variation of closed-subgroup theorem
See Tao's book on Hilbert's fifth problem -- <https://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/> -- Theorem 3.1.7, where it is referred to as local Cartan's theorem.


