Proposition 5

Let V be a vector space over k that is the direct sum of vector spaces $V_1, \ldots, V_r$. Let R be a root system in V. Put $R_i = R \cap V_i$. The following three conditions are equivalent:

(i) the $V_i$ are stable under $W(R)$; (ii) $R \subseteq V_1 \cup V_2 \cup \cdots \cup V_r$; (iii) for all $i$, $R_i$ is a root system in $V_i$, and R is the direct sum of the $R_i$. (iii) $\Longrightarrow$ (i): this follows from what we said at the beginning of this number.

(i) $\Longrightarrow$ (ii): assume that the $V_i$ are stable under $W(R)$. Let $\alpha \in R$ and let H be the kernel of $\alpha^-$. By Prop. 3 of Chap. V, § 2, no. 2, each $V_i$ is the sum of a subspace of H and a subspace of $k\alpha$. Hence one of the $V_i$ contains $k\alpha$, so $\alpha \in V_1 \cup V_2 \cup \cdots \cup V_r$.

(ii) $\Longrightarrow$ (iii): if condition (ii) is satisfied, $R_i$ generates $V_i$ for all $i$, so $R_i$ is a root system in $V_i$ (Prop. 4). It is clear that R is the direct sum of the $R_i$.