Proposition 6

Let $(\mathfrak{g},\mathfrak{h})$ be a split semi-simple Lie algebra, $\Phi$ a non-degenerate invariant symmetric bilinear form on $\mathfrak{g}$. For all $\alpha \in R$, let $X_{\alpha}$ be a non-zero element of $\mathfrak{g}^{\alpha}$. Let $(H_i)_{i\in I}$ be a basis of $\mathfrak{h}$, and $(H'_i)_{i\in I}$ the basis of $\mathfrak{h}$ such that $\Phi (H_i, H'_j) =\delta_{ij}$. The Casimir element associated to $\Phi$ in the enveloping algebra of $\mathfrak{g}$ (Chap. I, §3, no. 7) is then

$\alpha \in Ri\in I$

Indeed, by Prop. $1,\Phi (H_i, X_{\alpha}) =\Phi (H'_i, X_{\alpha}) = 0$ for all $i\in I,\alpha \in R$, and $\Phi (\frac{1}{\Phi(X_{\alpha},X_{-\alpha})}X_{\alpha}, X_{-\beta})=\delta_{\alpha \beta}$ for all $\alpha , \beta \in R$.