Proposition 12

Let $\mathscr{H}$ be the set of connected closed subgroups of G containing T, ordered by inclusion. The map $H \rightarrow R(H,T)$ is an increasing bijection from $\mathscr{H}$ to the set of symmetric closed subsets of $R(G,T)$, ordered by inclusion.

If $H\in \mathscr{H}$, then $L(H)_{(\mathbf{C})}$ is the direct sum of $\mathfrak{t}_{\mathbf{C}}$ and the $\mathfrak{g}^{\alpha}$ for $\alpha \in R(H,T)$; since this is a reductive subalgebra in $\mathfrak{g}_{\mathbf{C}}$, the subset $R(H,T)$ of R satisfies the stated conditions (Chap. VIII, §3, no. 1, Lemma 2 and Prop. 2). Conversely, if P is a subset of R satisfying these conditions, then $\mathfrak{t}_{\mathbf{C}}\oplus \sum_{\alpha\in P}\mathfrak{g}^{\alpha}$ is a subalgebra

of $\mathfrak{g}_{\mathbf{C}}($loc. cit.) which is rational over $\mathbf{R}$ (no. 3), and hence of the form $\mathfrak{h}_{(\mathbf{C})}$, where $\mathfrak{h}$ is a subalgebra of $\mathfrak{g}$. Let H(P) be the integral subgroup of G defined by $\mathfrak{h}$; it is closed (§2, no. 4, Remark 1). We verify immediately that the maps $H \rightarrow R(H,T)$ and $P \rightarrow H(P)$ are increasing and inverses of each other.