Corollary 3

Tag 0A05, permanent. Cite it as [Bourbaki, Tag 0A05].

Lie Groups and Lie Algebras, chapter I, § 6. Semi-simple Lie algebras

Let $g$ be a Lie algebra and $\rho$ and $\rho'$ two finite-dimensional semi-simple representations of $g$ on spaces $M$ and $M'$ . Then the representation of $g$ on $\mathscr{L}_K(M, M')$ canonically derived from $\rho$ and $\rho'$ is semi-simple.

The $g$ -module $\mathscr{L}_K(M, M')$ is canonically identified with the $g$ -module $M^* \otimes_K M'$ (§ 3, no. 3, Proposition 4), so that Corollary 3 follows from Corollary 1.

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

Cited by

Theorem 6, § 6
Proposition 1, § 5

Cites

§ 3, no. 3, Proposition 4
Corollary 1