On a Theorem of Wang for Complex Homogeneous Manifolds

In \cite{Wang1954}, Wang proved (among other things) a sufficiency result for a compact homogeneous manifold $G/H$ to admit a $G$-invariant complex structure.

In this note, we give a simple proof of Wang's theorem which relies on nothing more than the familiar properties of the root space decomposition of a compact Lie group.

It should be noted that the recent work of Ni and Wallach \cite{NiWallach2025} also revisits the aforementioned theorem of Wang (and others) and offers new Lie theoretic proofs as well.

However, the approach of \cite{NiWallach2025} relies on such objects as Borel subalgebras, parabolic subalgebras, and Iwasawa decomposition which may be somewhat less familiar to the working differential geometer.