Complexification of a real Lie algebra

Let $𝔲$ be a Lie algebra over $ℝ$ . The complexification $𝔤$ of $𝔲$ is the complexification of $𝔤$ as a vector space with a natural bracket, lie namely

𝔤 = ℂ \otimes ℝ 𝔲 = 𝔲 \oplus 𝑖 𝔲

and

[𝑧 \otimes 𝑥, 𝑤 \otimes 𝑦] = 𝑧 𝑤 \otimes [𝑥, 𝑦]

for $𝑧, 𝑤 \in ℂ$ and $𝑥, 𝑦 \in 𝔲$ . Hence $𝔤$ is the tensor product of a Lie algebra and a commutative algebra as well as an induced module.

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Complexification of a real Lie algebra

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