Corollary 4

There exists a Chevalley system $(X_{\alpha})_{\alpha\in R}$ in $(\mathfrak{g},\mathfrak{h}) ($§2, no. 4, Def. 3) such that $X_{\alpha}=x_{\alpha}$ and $X_{-\alpha}=y_{\alpha}$ for $\alpha \in B$. For every Chevalley system $(X'_{\alpha})_{\alpha\in R}$ having these properties, and for all $\alpha \in R,X'_{\alpha}$ is a basis of $\mathscr{G}\cap \mathfrak{g}^{\alpha}$.

For $\alpha \in B$, put $X_{\alpha}=x_{\alpha}, X_{-\alpha}=y_{\alpha}$. For $\alpha \in R_+$ -- B, choose a $w\in W$ such that $w(\alpha )\in B$ and an automorphism $\varphi$ of $\mathfrak{g}$ such that $\theta \varphi =\varphi \theta ,\varphi (\mathscr{G}) =\mathscr{G}$ and $\varphi (h) =w^{-1}(h)$ for $h\in \mathfrak{h}$ (Cor. 2); put $X_{\alpha}=\varphi (x_{w(\alpha)}), X_{-\alpha}=\varphi (y_{w(\alpha)})$. Then

so $(X_{\alpha})_{\alpha\in R}$ is a Chevalley system. Moreover,

Let $(X'_{\alpha})_{\alpha\in R}$ be a Chevalley system such that $X'_{\alpha}=x_{\alpha}, X'_{-\alpha}=y_{\alpha}$ for $\alpha \in B$. Let S be the set of $\alpha \in R$ such that $X'_{\alpha}=\pm X_{\alpha}$. By §2, no. 4, Prop. 7, S is a closed set of roots. Since $S\supset B\cup (-B)$, we have S = R (Chap. VI, §1, no. 6, Prop. 19). Thus, by (32) and (33), we have $\mathscr{G}\cap \mathfrak{g}^{\alpha}=\mathbf{Z}X'_{\alpha}$ for all $\alpha \in R$.