Remark

Let B be a basis of R, and $\mathfrak{b}$ the corresponding Borel subalgebra. If $\Sigma \subset B$, denote by $Q_{\Sigma}$ the set of roots that are linear combinations of elements of $\Sigma$ with coefficients $\leq 0$; put $\mathfrak{p}(\Sigma ) = R_+(B)\cup Q_{\Sigma}$ and $\mathfrak{p}_{\Sigma}=\mathfrak{h}\oplus \mathfrak{g}^{P(\Sigma)}$. By Chap. VI, §1, no. 7, Lemma 3 and Prop. $20,\mathfrak{p}_{\Sigma}$ is a parabolic subalgebra containing $\mathfrak{b}$ and every parabolic subalgebra of $\mathfrak{g}$ containing $\mathfrak{b}$ is obtained in this way.