Exercise 5, § 2

Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra, and $(X_{\alpha})_{\alpha\in R}$ a family of elements satisfying the conditions of Lemma 2. Let B be a basis of R. The $H_{\alpha}(\alpha \in B)$ and the $X_{\alpha}(\alpha \in R)$ form a basis of the vector space $\mathfrak{g}$. Show that the discriminant of the Killing form of $\mathfrak{g}$ with respect to this basis (Algebra, Chap. IX, §2) is a rational number, independent of the choice of $\mathfrak{h}$, of B, and of the $X_{\alpha}.^{10}$ Deduce that, if $n=$ dim $\mathfrak{g}$, the element of $\bigwedge^n\mathfrak{g}$ defined by the exterior product of the $H_{\alpha}(\alpha \in B)$ and the $X_{\alpha}(\alpha \in R)$ is independent, up to sign, of the choice of $\mathfrak{h}$, of B, and of the $X_{\alpha}$.