Proposition 24

Let $R'$ be the intersection of $R$ with a vector subspace of $V$, so that $R'$ is a root system in the vector subspace $V'$ that it generates (cf. Cor. of Prop. 4, no. 1). Let $B'$ be a basis of $R'$.

(i) There exists a basis of $R$ containing $B'$. (ii) $R'$ is the set of elements of $R$ that are linear combinations of elements of $B'$.

Assertion (ii) is clear. We prove (i). Let $(\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_l)$ be a basis of $V$ such that $B' = (\varepsilon_{p+1}, \varepsilon_{p+2}, \ldots, \varepsilon_l)$. The lexicographic order on $V$ corresponding to this basis defines a chamber $C$ of $R$. It is clear that every element of $B'$ is minimal in $R_+(C)$. Thus $B' \subseteq B(C)$.