Definition 2
Tag 0AEV, permanent. Cite it as [Bourbaki, Tag 0AEV].
Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups
Let a be a finite-dimensional Lie algebra. Let Ad(a) or Int(a) denote the integral subgroup of Aut(a) with Lie algebra ad(a). The elements of this group are called inner automorphisms of a.
By transport of structure, ad(a) is invariant under Aut(a) and hence Int(a) is normal in Aut(a). By § 4, no. 4, Corollary 1 to Proposition 8 and the fact that Int(a) is connected, the elements of Int(a) are the finite products of automorphisms of the form exp ad x where x ∈ a. In general, Int(a) is not a Lie subgroup of Aut(a) (Exercise 14).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.