Connected subgroup determined by its Lie algebra

Let $G$ be a Lie group . The slogan “Lie algebra determines the connected subgroup” can be made precise as follows.

Theorem (uniqueness)

If $H_1,H_2\subseteq G$ are connected Lie subgroups and

then $H_1=H_2$.

Equivalently: for any Lie subalgebra $\mathfrak h\subseteq \mathfrak g$, there is at most one connected Lie subgroup $H\subseteq G$ with $\mathrm{Lie}(H)=\mathfrak h$.

Why this is true

One convenient way to see the mechanism is via the exponential map . For a connected Lie subgroup $H$ with Lie algebra $\mathfrak h$, the subgroup $H$ is generated by exponentials $\exp(\mathfrak h)$ (more precisely, by $\exp(U\cap \mathfrak h)$ for any sufficiently small neighborhood $U$ of $0$). Thus if two connected subgroups share $\mathfrak h$, they are generated by the same one-parameter subgroups and must coincide.

This uniqueness is one half of the Lie correspondence between connected subgroups and subalgebras; existence typically uses integrability of left-invariant distributions or the third theorem of Lie .