Remark

Tag 09V4, permanent. Cite it as [Bourbaki, Tag 09V4].

Lie Groups and Lie Algebras, chapter I, § 2. Enveloping algebra of a Lie algebra

Let g' be another Lie algebra over the ring K, U' its enveloping algebra, U'_n the set of elements of U' of filtration $\leq n$ and U^n (resp. U'^n) the set of canonical images in U (resp. U') of the homogeneous symmetric tensors of g (resp. g') of order n. Let $\eta$ be a homomorphism of g into g' and let $\tilde{\eta}$ be the corresponding homomorphism of U into U'. Then

\tilde{\eta}(U_n) \subset U'_n, \quad \tilde{\eta}(U^n) \subset {U'}^n.

In particular, the principal antiautomorphism of U leaves U_n and U^n stable. The K-linear mapping of T^n onto itself which maps

x_1 \otimes x_2 \otimes \cdots \otimes x_n \quad \text{to} \quad x_n \otimes x_{n-1} \otimes \cdots \otimes x_1

for all $x_1, ..., x_n$ in g is a symmetry operator and hence leaves fixed the homogeneous symmetric tensors of order n. Hence the principal antiautomorphism of U induces on each U^n the homothety of ratio $(-1)^n$ .

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

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