Let $G$ be a subgroup of $GL_n(\mathbb{C})$, and $\mathfrak{g}$ be its Lie algebra. In other words, $\mathfrak{g}=T_I G$. We know if $X\in \mathfrak{g}$, then $\exp(tX)\in G$ for any $t\in \mathbb{R}$.
I am wondering whether another direction is correct, namely:
if $\exp(tX)\in G$ for any $t\in \mathbb{R}$, then $X\in \mathfrak{g}$.
It sounds correct, and I actually need it to prove a statement on the intersection of Lie algebras. Could anyone help me check whether the above direction is correct or not? If this direction is actually incorrect, can anyone give me a hint on how to prove the statement in the link?
You just have to observe that $t\mapsto\exp(tX)$ is a smooth curve in $M_n(\mathbb R)$ and its derivative at $0$ is $X$. So if the curve has values in $G$, then by definition its derivative at $0$ lies in $\mathfrak g$. (So it would be sufficient, that the values lie in $G$ in some neighborhood of $0$.)