Corollary

Tag 0A47, permanent. Cite it as [Bourbaki, Tag 0A47].

Lie Groups and Lie Algebras, chapter II, § 3. Enveloping algebra of the free Lie algebra

Suppose that K is a Q-algebra. Let P be the linear mapping of $A^+(X)$ into itself such that

P(x_1 \ldots x_n) = \frac{1}{n} (\operatorname{ad} x_1 \circ \cdots \circ \operatorname{ad} x_{n-1})(x_n)

for $n \geq 1$ and $x_1, \ldots, x_n$ in X. Then P is a projector of $A^+(X)$ onto $L(X)$ .

The image of P is contained in $L(X)$ . Further, for all $n \geq 1$ and all $u$ in $L^n(X)$ , $P(u) = \frac{1}{n} \pi(u)$ , whence $P(u) = u$ by Proposition 1. As

L(X) = \sum_{n \geq 1} L^n(X),

we see that the restriction of P to $L(X)$ is the identity.

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

Cited by

Theorem 2, § 6

Cites

Proposition 1