This is related to an Lemma 7.34 of Brian Hall, Lie group, Lie Algebra and Representations chpt 7.

Lemma 7.34: Let $K$ be a compact matrix lie group with non-commutative lie algebra. Then real lie algebra $k$ of $K$ cannot have complex structure(i.e. $\phi:g\to k$ is a lie algebra homomorphism with $g=l_C$ as complexification of some real lie algebra $l$ coming from some reductive lie group $L$. $\phi\circ(i\cdot)\circ\phi^{-1}$ induces complex multiplication on $k$.)

Since this complex structure does not say integrability condition, I would call this complex structure as almost complex structure.

$\textbf{Q:}$ It seems that in order to have complex structure on lie algebra, I need either commutative lie algebra or non-compact matrix lie group. What will guarantee existence of complex structure on a lie algebra? I would expect this complex structure pulls back by exponential to lie group and generate global trivialization of tangent bundle via left or right invariant vector field.(In particular, this will result in global splitting of complexified tangent bundle of lie group.)

$\textbf{Q':}$ Is there any intuitive reason to expect for compact non-commutative lie group's lie algebra not having the complex structure?

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    $\begingroup$ See also this question. $\endgroup$ – Dietrich Burde Feb 20 at 17:58
  • $\begingroup$ This is purely algebraic, so there's no meaningful integrability condition and no reason to call this almost complex instead of complex. At the Lie group level however, it makes sense, and if a real Lie group has a complex structure on its Lie algebra, then it indeed induces a complex structure on the Lie group (i.e., the almost complex structure it induces is integrable). $\endgroup$ – YCor Feb 21 at 1:15
  • $\begingroup$ As regards Q', "intuitive" depends on your background. There are both arguments using complex analysis, and also purely algebraic at the Lie algebra level. $\endgroup$ – YCor Feb 21 at 1:18
  • $\begingroup$ @YCor What is the "intuitive" arguments for both complex analysis and algebraic level argument? Thanks. $\endgroup$ – user45765 Feb 21 at 3:46

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