# If $H \subset G$ is a connected Lie subgroup of the same dimension, does it imply that $H = G$?

Let $$G$$ be a real (finite-dimensional) connected Lie group. Suppose that $$H \subset G$$ is a connected Lie subgroup, and $$\dim(H) = \dim(G)$$, does it imply that $$H = G$$?

If not, what would be a simple counterexample? Any mild additional assumption would make this hold? Thanks!

Note that since $$dim(H) = dim(G)$$ the inclusion $$H \hookrightarrow G$$ is an open map, therefore $$H$$ is open in $$G$$. We know that the identity is in $$H$$, but for any open set containing the identity, the group genrated by it is the whole group (G is connected), thus $$H=G$$.