Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$ and suppose that $K$ is a compact connected subgroup of $G$ with Lie algebra $\mathfrak{k}$ such that $\mathfrak{g}=\mathfrak{k}\otimes\mathbb{C}$ is the complexification of $\mathfrak{k}$.
Does this imply that $G$ is the complexification of $K$?
Here by complexification, I mean this:
Definition. A complex Lie group $G$ is a complexification of a Lie group $K\subseteq G$ if every Lie group homomorphism $K\to H$ where $H$ is a complex Lie group extends to a unique holomorphic Lie group homomorphism $G\to H$. (See Wikipedia.)
Otherwise said, I want to know if $G$ is a complex reductive group in the sense of algebraic geometry (so that we can consider GIT quotients).
It is known that if $K$ is a compact Lie group, then there is a unique complex Lie group $G$ which is the complexification of $K$. Moreover, in that case $\mathfrak{g}=\mathfrak{k}\otimes\mathbb{C}$.