I need someone to review my following work:
Let $G$ be a connected abelian complex Lie group and let $K$ be a maximal compact subgroup of $G$. Let $\mathfrak g$ and $\mathfrak k $ be the Lie algebras of $G$ and $K$ resp. such that $\mathfrak g=\mathfrak k+i\mathfrak k$.
(I am not sure about the following) We can write $\mathfrak g=(\mathfrak k+i\mathfrak k)/(\mathfrak k\cap i\mathfrak k)\oplus \mathfrak k\cap i\mathfrak k$. We also have,
- $(\mathfrak k+i\mathfrak k)/(\mathfrak k\cap i\mathfrak k)$ is totally real.
- We can write $G$ as a direct sum of two abelian complex Lie groups so that the Lie algebra of the first group is $(\mathfrak k+i\mathfrak k)/(\mathfrak k\cap i\mathfrak k)$ and the Lie algebra for the second group is $\mathfrak k\cap i\mathfrak k$.