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.
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David N. Pham (2024) studied this question.
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