Corollary 1

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

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let G be a Lie group, of finite dimension if K is of characteristic > 0, and $\omega$ and $\omega'$ the left and right canonical differential forms of G. Then

d\omega + [\omega]^2 = 0, \qquad d\omega' - [\omega']^2 = 0.

By Proposition 51,

(d\omega)_e(a_1, a_2) = -\omega_e([a_1, a_2]) = -[a_1, a_2] = -[\omega_e(a_1), \omega_e(a_2)] = -[\omega]^2_e(a_1, a_2)

whence the first formula. The second can be established analogously.

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

Cited by

Proposition 60, § 3

Cites

Proposition 51