Corollary 3
Tag 09YP, permanent. Cite it as [Bourbaki, Tag 09YP].
Lie Groups and Lie Algebras, chapter I, § 5. Solvable Lie algebras
Let g be a Lie algebra, r its radical and a an ideal of g. Then the radical of a is equal to .
r ∩ a is a solvable ideal of a and hence is contained in the radical r' of a. Conversely, r' is an ideal of g (Corollary 2 and § 1, no. 4, Proposition 2) and hence r' ⊂ r.
Corollary 2 can be made more precise as follows:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.