Proposition 13

Tag 0A9P, permanent. Cite it as [Bourbaki, Tag 0A9P].

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

Let $t \in \mathscr{T}^{(s)}(G)$ , $t' \in \mathscr{T}^{(s')}(G)$ , $u \in \mathscr{T}^{(s'')}(X)$ , such that

s + s' + s'' \leq r.

Then $(t * t') * u = t * (t' * u)$ .

This can be proved as is Proposition 2 of no. 1.

In particular, if $r \leq \infty$ , the vector space $\mathscr{T}^{(\infty)}(X)$ is a left module over the algebra $\mathscr{T}^{(\infty)}(G)$ with the product *.

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

Cited by

§ 3. Passage from a Lie group to its Lie algebra

Cites

Proposition 2