Exercise 1, § 1

Tag 00YE, permanent. Cite it as [Bourbaki, Tag 00YE].

Lie Groups and Lie Algebras, VIII, 219

Let U be the enveloping algebra of $\mathfrak{s}$ . Show that the element

C =H^2-2(X_+X_-+X_-X_+) =H^2+ 2H-4X_-X_+

belongs to the centre of U, and that its image in the representation associated to $V(m)$ is the homothety with ratio $m(m+ 2)$ .

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