Proposition 2

Let $\alpha , \beta \in R$.

(i) $\beta (H_{\alpha})\in \mathbf{Z}$.

(ii) If $\Phi$ denotes the Killing form of $\mathfrak{g},\Phi (H_{\alpha}, H_{\beta})\in \mathbf{Z}$.

Let $X_{\alpha}$ be a non-zero element of $\mathfrak{g}^{\alpha}$, and let $\rho$ be the representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$ associated to $X_{\alpha}$. The eigenvalues of $\rho (H)$ are 0 and the $\beta (H_{\alpha})$ for $\beta \in R$. Hence (i) follows from §1, no. 2, Cor. of Prop. 2. Assertion (ii) follows from (i) and the Cor. of Th. 1. Q.E.D.

Let $\alpha \in R,X_{\alpha}$ a non-zero element of $\mathfrak{g}^{\alpha},X_{-\alpha}$ the element of $\mathfrak{g}^{-\alpha}$ such that $[X_{\alpha}, X_{-\alpha}] =-H_{\alpha}$, and $\rho$ the representation of $\mathfrak{s}\mathfrak{l}(2, k)$ on $\mathfrak{g}$ associated to $X_{\alpha}$. Let $\pi$ be the representation of $\mathbf{S}\mathbf{L}(2, k)$ on $\mathfrak{g}$ compatible with $\rho ($§1, no. 4, Th. 2). Since ad $X_{\alpha}$ is nilpotent (Chap. VII, §1, no. 3, Prop. 10 (iv)), $\pi (e^{X_+}) =e^{adX_{\alpha}}$ is an elementary automorphism of $\mathfrak{g}$. Similarly, $\pi (e^{X_-}) =$ $e^{adX_{-\alpha}}$ is an elementary automorphism of $\mathfrak{g}$. Hence $\pi (\mathbf{S}\mathbf{L}(2, k))\subset$ Aut$_e(\mathfrak{g})$. We make use of the notation $\theta (t)$ of §1, no. 5. For $t\in k^*$, put