Corollary 2
Tag 09ZG, permanent. Cite it as [Bourbaki, Tag 09ZG].
Lie Groups and Lie Algebras, chapter I, § 6. Semi-simple Lie algebras
Every extension of a semi-simple Lie algebra by a semi-simple Lie algebra is semi-simple and trivial.
This follows immediately from Corollary 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.