TITLE: Does every even-dimensional Lie group admit a complex structure?
QUESTION [10 upvotes]: Let $M$ be an even-dimensional Lie group. Is it true that $M$ admits a complex structure?
I can see that there exists an almost complex structure, since the tangent bundle is trivial. But this is not sufficient Trivality of tangent bundle and complex structure.
So I'm wondering if adding the Lie group structure is enough to guarantee integrable. Maybe the Nijenhuis tensor simplifies in some way?

REPLY [6 votes]: In the case of compact Lie groups, it is true that an even-dimensional Lie group always admits a complex structure. See Example 4.32 in Félix, Oprea, Tanré, Algebraic Models in Geometry (available here https://www.maths.ed.ac.uk/~v1ranick/papers/tanre.pdf).