Corollary 5

Let B be a basis of R, $A_1, A_2, \ldots, A_r$ pairwise orthogonal subsets of B, and $A = A_1 \cup A_2 \cup \cdots \cup A_r$. Then every root $\alpha$ that is a linear combination of elements of A is actually a linear combination of elements of one of the $A_i$. In particular, if R is irreducible, there is no partition of B into pairwise orthogonal subsets.

Let $E_1, \ldots, E_r, E$ be the vector subspaces of V generated by $A_1, \ldots, A_r, A$, respectively. By Cor. 4, we can assume that $E = V$. Then, by Th. 2 (vii) of no. 5, the $E_i$ are stable under $W(R)$, so R is the union of the $R \cap E_i$ (no. 2, Prop. 5).