I am trying to understand the statement of Satz 1 in Über kompakte homogene Kählersche Mannigfaltigkeiten by Borel. Here is the statement in German

Satz I: Jede zusammenhängende kompakte homogene Kählersche Mannigfaltigkeit ist das direkte Produkt aus einem komplexen Torus und einer projektiv-rationalen Mannigfaltigkeit.

I think this states that a compact homogeneous Kähler manifold is a product of a complex torus and a projective rational manifold. Now surely the projective rational manifold should also be homogeneous? Is this somehow implied? If this is the case then does that mean that projective rational manifold is in fact a complex flag manifold?