Derivation on an algebra

Inner derivation

Let 𝔤 be a Lie algebra and ad(−) be its adjoint representation. An inner derivation is a derivation on 𝔤 that can be written as ad𝑥 for some 𝑥 ∈𝔤. The set of all such derivations 𝔦𝔫𝔫⁡(𝔤) is an ideal of [[Derivation subalgebra|𝔡𝔢𝔯⁡(𝔤)]].

Proof of ideal

Let 𝐷 ∈𝔡𝔢𝔯⁡(𝔤) be a derivation, and let ad𝑥 ∈𝔡𝔢𝔯⁡(𝔤). Then for any 𝑦 ∈𝔤

[𝐷,ad𝑥]𝑦=𝐷ad𝑥⁡𝑦−ad𝑥⁡𝐷𝑦=𝐷[𝑥,𝑦]−[𝑥,𝐷𝑦]=[𝐷𝑥,𝑦]+[𝑥,𝐷𝑦]−[𝑥,𝐷𝑦]=ad𝐷𝑥⁡𝑦

as required.

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