[[Lie algebra automorphism]]
# Inner Lie algebra automorphism

An **inner automorphism** of a [[Lie algebra]] $\mathfrak{g}$ is given by $\mathrm{e}^{-\ad_{x}}$, where $\ad_{x}$ is the [[Adjoint Lie algebra representation|adjoint representation]] for some $x \in \mathfrak{g}$. #m/def/lie 
As the [[Exponential of a derivation on a Lie algebra]] it is an [[Lie algebra automorphism|automorphism]].[^1972]
The set of all inner automorphisms $\Inn \mathfrak{g}$ forms a [[normal subgroup]] of $\Aut \mathfrak{g}$.

> [!missing]- Proof of normal subgroup
> #missing/proof

  [^1972]: 1972\. [[Sources/@humphreysIntroductionLieAlgebras1972|Introduction to Lie Algebras and Representation Theory]], §2.3, pp. 8–9

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