Let $G$ be a complex Lie group and $H$ be a closed subgroup and let $J$ be a algebraic subgroup of $G$. If $G/J$ is an quasi-affine. Then why the holomorphic canonical map $\pi: G/H \rightarrow G/J$ splits as follows
\begin{array}{ccccccccc} G/H & \xrightarrow{\pi} & G/J & \\ \searrow & &\nearrow \\ & G/\bar H \end{array}Where $\bar H$ is the Zariski-closure of $H$?