Proposition 5

Tag 09W6, permanent. Cite it as [Bourbaki, Tag 09W6].

Lie Groups and Lie Algebras, chapter I, § 3. Representations

Let g be a Lie K-algebra, h an ideal of g, $\rho$ a representation of g on M and $\rho'$ the restriction of $\rho$ to h. Then the set N of elements of M invariant with respect to $\rho'$ is stable under $\rho(g)$ .

Let $n \in N$ and $y \in g$ ; for all $x \in h, [x, y] \in h$ and hence

\rho(x)\rho(y)n = \rho([x, y])n + \rho(y)\rho(x)n = 0;

hence

\rho(y)n \in N

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

Cited by

Lemma 2, § 4
Theorem 1 (Engel), § 4
Proposition 8, § 5

Cites

Nothing.