Corollary
Tag 0ANW, permanent. Cite it as [Bourbaki, Tag 0ANW].
Lie Groups and Lie Algebras, chapter IV, § 2. Tits Systems
Under the assumptions of Th. 5, the group G_1/(G_1 \cap Z) is either simple non-abelian or reduces to the identity element.
Th. 5 shows that G_1/(G_1 \cap Z) is simple or reduces to the identity element. On the other hand, assumption (3) implies that it is equal to its commutator subgroup. Hence the corollary.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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