Corollary 3

Tag 0AFI, permanent. Cite it as [Bourbaki, Tag 0AFI].

Lie Groups and Lie Algebras, chapter III, § 6. Real and complex Lie groups

Let $x \in E$ . For $x$ to be invariant under $\varphi(G)$ , it is necessary and sufficient that $x$ be annihilated by $L(\varphi)(L(G))$ (that is that $x$ be invariant under $L(\varphi)$ in the sense of Chapter I, § 3, Definition 3).

It suffices to take $E_1 = Kx$ and $E_2 = 0$ in Proposition 13.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 4, § 6
Corollary 6, § 6
§ 1. Compact Lie algebras
Proposition 1, § 7

Cites

Chapter I, § 3, Definition 3
Proposition 13