Remark 4

Let $\mathfrak{g}$ be a splittable semi-simple Lie algebra, $\mathscr{P}$ the set of parabolic subalgebras of $\mathfrak{g}$, a set on which Aut$_0(\mathfrak{g})$ operates. Retain the notations of Remark 2. Let $\Sigma$ be a subset of $B(\mathfrak{g})$. Giving $\Sigma$ is equivalent to giving, for every $x= (\mathfrak{h},B)\in X$, a subset $\Sigma_x$ of B, such that $s_{x',x}$ takes $\Sigma_x$ to $\Sigma_{x'}$ for any $x, x'\in X$. Let $\mathfrak{p}_x$ be the parabolic subalgebra of $\mathfrak{g}$ corresponding to $\Sigma_x$ (§3, no. 4, Remark). The orbit of $\mathfrak{p}_x$ under Aut$_0(\mathfrak{g})$ is the set of $\mathfrak{p}_{x'}$ for $x'\in X$. This defines a map from $\mathfrak{P}(B(\mathfrak{g}))$ to $\mathscr{P}/$Aut$_0(\mathfrak{g})$. This map is surjective by the Remark of §3, no. 4, and injective by Prop. 7.