Let $K$ be an abelian compact Lie group (torus) and let $K^\mathbb C$ be its complexification. Let $\frak k$ be the Lie algebra of $K$ and $\mathfrak k^\mathbb C=\mathfrak k+i\frak k$ be the Lie algebra of $K^\mathbb C$.
Let $\frak s$ be a Lie subalgebra of $\mathfrak k^\mathbb C$ isomorphic to $\mathfrak k^\mathbb C/(\mathfrak k\cap i\frak k)$.
I am trying to show that $\overline{exp(\mathfrak s)}\cong\overline {exp (\mathfrak k^\mathbb C/(\mathfrak k\cap i\frak k))}\cong K^\mathbb C/\overline {exp(\mathfrak k\cap i\frak k)}\cong (\mathbb C^*)^n$? But I am not sure how to do that? Is it even correct that $\overline {exp (\mathfrak k^\mathbb C/(\mathfrak k\cap i\frak k))}\cong(\mathbb C^*)^n$?