Proposition 8

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

Lie Groups and Lie Algebras, chapter I, § 3. Representations

Let $g$ be a Lie algebra and $M$ a $g$ -module. Suppose that $M$ , considered as a $K$ -module, admits a finite basis. The bilinear form associated with $M$ is invariant.

For $x, y, z$ in $g$ , we have:

\begin{align*} \operatorname{Tr}([x, y]_M z_M) &= \operatorname{Tr}(x_M y_M z_M) - \operatorname{Tr}(y_M x_M z_M) = \operatorname{Tr}(x_M y_M z_M) - \operatorname{Tr}(x_M z_M y_M) \\ &= \operatorname{Tr}(x_M [y, z]_M). \end{align*}

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

Cited by

Nothing.

Cites

Nothing.