Corollary 5

Tag 0AB9, permanent. Cite it as [Bourbaki, Tag 0AB9].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let $A$ be a finite-dimensional algebra. Suppose that $K$ is of characteristic 0. The automorphism group $\mathrm{Aut}(A)$ of $A$ is a Lie subgroup of $\mathbf{GL}(A)$ and $L(\mathrm{Aut}(A))$ is the Lie algebra of derivation of $A$ .

This follows from Corollary 1 (applied to $E = \mathscr{L}(A, A; A)$ ) and Corollary 2 of Proposition 39 (applied to the subset of $E$ consisting only of multiplication on $A$ ).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 6

Cites

Corollary 2 of Proposition 39
Corollary 1