Sum of commuting Lie algebra representations

Let $𝔤$ be a Lie algebra over $𝕂$ , and $𝜋 1, 𝜋 2 : 𝔤 \to 𝔤 𝔩 (𝑉)$ be representations of $𝔤$ on a $𝕂$ -linear space $𝑉$ that commute in the sense that for any $𝑥, 𝑦 \in 𝔤$

[𝜋 1 (𝑥), 𝜋 2 (𝑦)] = 0

Then $𝜋 = 𝜋 1 + 𝜋 2$ is a representation of $𝔤$ on $𝑉$ . #m/thm/lie

Proof

Since

𝜋 ([𝑥, 𝑦]) = 𝜋 1 ([𝑥, 𝑦]) + 𝜋 2 ([𝑥, 𝑦]) = 𝜋 1 (𝑥) 𝜋 1 (𝑦) - 𝜋 1 (𝑦) 𝜋 1 (𝑥) + 𝜋 2 (𝑥) 𝜋 2 (𝑦) - 𝜋 2 (𝑦) 𝜋 2 (𝑥) = 𝜋 1 (𝑥) 𝜋 1 (𝑦) + 𝜋 2 (𝑥) 𝜋 2 (𝑦) - 𝜋 1 (𝑦) 𝜋 1 (𝑥) - 𝜋 2 (𝑦) 𝜋 2 (𝑥) . + 𝜋 1 (𝑥) 𝜋 2 (𝑦) + 𝜋 2 (𝑥) 𝜋 1 (𝑦) - 𝜋 2 (𝑦) 𝜋 1 (𝑥) - 𝜋 1 (𝑦) 𝜋 2 (𝑥) = 𝜋 (𝑥) 𝜋 (𝑦) - 𝜋 (𝑦) 𝜋 (𝑥)

as required.

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