Definition 1
Tag 019E, permanent. Cite it as [Bourbaki, Tag 019E].
Lie Groups and Lie Algebras, chapter IX, § 1. Compact Lie algebras
A compact Lie algebra is a Lie algebra that has properties (i) to (v) of Proposition 1.
Thus, the compact Lie algebras are the products of a commutative algebra with a compact semi-simple algebra. In other words, a Lie algebra is compact if and only if it is reductive and its derived Lie algebra is compact.
The Lie algebra of a compact Lie group is compact.
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.