Definition 2
Tag 012P, permanent. Cite it as [Bourbaki, Tag 012P].
Lie Groups and Lie Algebras, chapter VIII, § 6. Modules over a split semi-simple Lie algebra
The homomorphism of Prop. 1 (iv) is called the central character of the -module V.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.