Proposition 23

Let $P$ be a subset of $R$ and $V_1$ (resp. $\Gamma$) the vector subspace (resp. the subgroup) of $V$ generated by $P$. The following conditions are equivalent:

(i) P is closed and symmetric; (ii) P is closed, and P is a root system in $V_1$; (iii) $\Gamma \cap R = P$.

Assume that these conditions are satisfied. For any $\alpha \in P$, let $\alpha_1 \tilde{}$ be the restriction of $\alpha \tilde{}$ to $V_1$. Then the map $\alpha \mapsto \alpha_1 \tilde{}$ is the canonical bijection from the root system $P$ to $P \tilde{}$.

(iii) $\Longrightarrow$ (i): clear.

(i) $\Longrightarrow$ (ii): assume that $P$ is closed and symmetric. First, $P$ satisfies (RS_I) in $V_1$. We show that, if $\alpha, \beta \in P$, then $s_\alpha(\beta) \in P$. This is clear if $\alpha$ and $\beta$ are proportional. Otherwise, $s_\alpha(\beta) = \beta - n(\beta, \alpha)\alpha$ and $\beta - p\alpha \in R$ for all rational integers $p$ between 0 and $n(\beta, \alpha)$ (Prop. 9, no. 3), so

since $P$ is closed and symmetric. Thus, $s_{\alpha, \alpha_1}(P) = P$, and $P$ satisfies (RS_{II}). It is clear that $P$ satisfies (RS_{III}). Thus, $P$ satisfies (ii), and we have proved the last assertion of the proposition at the same time.

(ii) $\Longrightarrow$ (iii): we show that, if condition (ii) is satisfied, then $\Gamma \cap R = P$. It is clear that $P \subseteq \Gamma \cap R$. Let $\beta \in \Gamma \cap R$. Since $\beta \in \Gamma$ and $P = -P$, $\beta = \alpha_1 + \alpha_2 + \cdots + \alpha_k$ with $\alpha_1, \ldots, \alpha_k \in P$. We shall prove that $\beta \in P$. This is clear if $k = 1$. We argue by induction on $k$. We have

so $(\beta|\alpha_i) > 0$ for some index $i$. If $\beta = \alpha_i$, then $\beta \in P$. Otherwise, $\beta - \alpha_i \in R$ (Cor. of Th. 1, no. 3), so $\beta - \alpha_i \in P$ by the induction hypothesis, hence $\beta \in P$ since $P$ is closed.

The conditions of Prop. 23 can be realised with $V_1 = V$ and yet $P \neq R$. For example, this is the case when $R$ is a system of type $G_2$ and $P$ a system of type $A_2$; cf. Plate X.