Let $G$ be a finite dimensional Lie group. Suppose there is a point $g$ such that there exists two tangent vectors $X,Y\in T_eG$ with $X\neq Y$ and $\exp(X)=exp(Y)=g$. In other words, the group exponential is not injective. Does it tell us something about a compact part of $G$, like: does it imply that $G$ has a compact subgroup, or that there is a morphism from $G$ to a non trivial compact group? I would not be surprised if it was the case, but I am not able to formalise it or to find appropriate references.
Thank you for your help.
