Corollary 3

A subset B of R is a basis of R if and only if the following conditions are satisfied:

(i) the elements of B are linearly independent; (ii) every root of R is a linear combination of elements of B in which the coefficients are either all positive or all negative; (iii) every root of B is indivisible.

We already know that the conditions are necessary (no. 5, Th. 2, and no. 6, Th. 3). Assume that conditions (i), (ii), (iii) are satisfied. Let P be the set of roots that are linear combinations of elements of B with coefficients $\geq 0$. Since P satisfies condition (ii) of Cor. 1, there exists a chamber C such that $P = R_+(C)$; let $B' = B(C)$, and let X and $X'$ be the convex cones generated by B and $B'$. Then

which shows that X and $X'$ are both generated by P, and hence coincide. But the half-lines generated by the elements of B (resp. by $B'$) are the extreme generators of X (resp. $X'$); since such a half-line contains only one indivisible root, $B = B'$.