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Let $\mathfrak{g}$ be a Lie algebra. Then its automorphism group $Aut(\mathfrak{g})$ is a Lie group, and hence we may take its Lie algebra $Lie(Aut(\mathfrak{g}))$.

I'd like to say that this is equal to the Lie algebra of derivations $Der(\mathfrak{g})$. Is this true? Where can I find a reference?

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  • $\begingroup$ Yes, it is true, see here. $\endgroup$ – Dietrich Burde Jun 1 '18 at 8:25
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This is a standard fact from the theory of Lie algebras. There are several references, say over $\Bbb{R}$ and $\Bbb{C}$, e.g., Proposition 1.25 in the book The Structure of Complex Lie Groups.

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