Consider the compact Lie group $G=T\cdot S$ with finite intersection $I:=T\cap S$, where $T$ is the central torus and $S$ the maximal semisimple subgroup. Now, $S/I\cong G/T$ has the structure of a complex flag manifold.
Since $S/I$ is a Lie group, Does it mean that it has the structure of a complex Lie group? I was thinking in the case where the intersection is finite then if $S$ is complex Lie group then it is abelian since it is compact!
So if $S/I$ is not a complex Lie group, what its complexification?