Remark
Tag 09YS, permanent. Cite it as [Bourbaki, Tag 09YS].
Lie Groups and Lie Algebras, chapter I, § 5. Solvable Lie algebras
To summarize some of the above results, note that, if r, n, s, t denote respectively the radical of g, the largest nilpotent ideal of g, the nilpotent radical of g and the orthogonal of g with respect to the Killing form, then
The inclusion r ⊃ t follows from Proposition 5 (b). The inclusion t ⊃ n follows from § 4, no. 4, Proposition 6 (b). The inclusion n ⊃ s has been pointed out in Remark 2 of no. 3.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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