What is the Lie group of $\mathfrak{h}$?

Question

Let $\mathfrak{h}$ be a Cartan subalgebra of a Lie group $G$. What is the Lie group of $\mathfrak{h}$? By definition, the Lie group of $\mathfrak{h}$ consisting of elements of the form $e^{h}$, $h \in \mathfrak{h}$. Is the Lie group of $\mathfrak{h}$ isomorphic to $\mathfrak{h}^*$? Thank you very much.

Answer 1

A Cartan subalgebra $\mathfrak{h}$ is a nilpotent and self-normalising subalgebra of the Lie algebra $\mathfrak{g}=Lie(G)$. Hence $H=\exp(\mathfrak{h})$ is a nilpotent Lie group. It need not be commutative in general. If $G$ is nilpotent, then $H=G$, because a nilpotent Lie algebra has only one Cartan subalgebra - itself.