Exercise 11, § 2

Let $\mathfrak{h}$ be a commutative Lie algebra $\not= 0, P$ a finite subset of $\mathfrak{h}^*$ containing 0. Show that there exists a Lie algebra $\mathfrak{g}$ containing $\mathfrak{h}$ as a Cartan subalgebra, and such that the set of weights of $\mathfrak{h}$ in $\mathfrak{g}$ is P. (Construct $\mathfrak{g}$ as the semi-direct product of $\mathfrak{h}$ by the $\mathfrak{h}$-module V which is the direct sum of the 1-dimensional modules corresponding to the elements of P -- $\{0\}$, cf. Exerc. 7.)

An element $x$ of $\mathfrak{h}$ is such that $\mathfrak{h}=\mathfrak{g}^0(x)$ if and only if $x$ is not orthogonal to any element of P-- $\{0\}$.