What is the name of the theorem that says that ${\frak{g}} \subseteq \{X \in M_n(R) \mid \forall t \in R : e^{tX} \in G\}$? For a matrix Lie group $G \subseteq GL_n(\mathbb{R})$, we define the Lie algebra $\frak{g}$ as $T_0 G$. We then proceed to prove the inclusion in the title to give a definition of $\frak{g}$ in terms of the exponential map.
I am very interested in learning if this theorem has a name.
 A: More generally, if $G$ is a Lie subgroup of a Lie group $H$ with respective Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, then
$$\mathfrak{g}=\{X\in\mathfrak{h}:\exp(tX)\in G,\forall t\in\mathbb{R}\}.$$
I have never seen any name attached to that result, since it is quite an elementary one. (It ultimately follows from uniqueness of solutions to ODEs.) Any name attached to it would not be very standard anyway. But in any case, if you use this result, you will much more understood if you clearly state it in general form  and give a reference (for example, Lee, Introduction to smooth manifolds, Second Edition, Proposition 20.9). And perhaps briefly explain how it implies your particular example: It is because the Lie algebra of $\mathrm{GL}(n,\mathbb{R})$ is isomorphic to $\mathrm{Mat}(n,\mathbb{R})$ and the exponential map of $\mathrm{GL}(n,\mathbb{R})$ is given by the matrix exponential $\mathrm{Mat}(n,\mathbb{R})\to \mathrm{GL}(n,\mathbb{R}):X\mapsto e^X$. (See the reference above for a proof of this fact.)
