Samelson complex structures for the tangent Lie group

Abstract

It is shown that for any compact Lie group $G$ (odd or even dimensional), the tangent bundle $TG$ admits a left-invariant integrable almost complex structure, where the Lie group structure on $TG$ is the natural one induced from $G$. The aforementioned complex structure on $TG$ is inspired by Samelson's construction for even dimensional compact Lie groups.