Proposition 4

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

Lie Groups and Lie Algebras, chapter I, § 1. Definition of Lie algebras

Let g and h be two Lie algebras over K and f a homomorphism of g onto h. Then $f(\mathscr{D}^pg) = \mathscr{D}^pf, f(\mathscr{C}^pg) = \mathscr{C}^ph$ .

If a and b are submodules of g, it follows immediately that

f([a, b]) = [f(a), f(b)].

The proposition is then immediate by induction on

p

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

Cited by

Corollary, § 1
Exercise 4, § 4
Corollary, § 2
Theorem 3, § 3

Cites

Nothing.