What am I misunderstanding about Lie groups?

Question

There is a theorem stating that there is a one-to-one correspondence between connected Lie subgroups of a Lie group and subalgebras of its Lie algebra.

Note that $GL(n,\mathbb{R})$ is a closed Lie subgroup of $GL(n,\mathbb{C})$ and $gl(n,\mathbb{R})$ is a Lie subalgebra of $gl(n,\mathbb{C})$. By the above theorem, $GL(n,\mathbb{R})$ must be connected, but it is not.

What am I misunderstanding here?

Answer 1

The Lie agebra of $Gl(n,\mathbb{R})$ is also the Lie algebra of its connected component which is the subset of elements with positive determinant.