Corollary 1

Let $\alpha \in R_+(C)$. Then $\alpha \in B(C)$ if and only if $\alpha$ is the sum of two positive roots.

If $\alpha$ is the sum of two positive roots, Th. 3 shows that $\alpha \in B(C)$. If $\alpha \notin B(C)$, Th. 3 shows that $\alpha = \sum_{k=1}^n \beta_k$ with $\beta_k \in B(C)$ for all $k$ and $n \geq 2$.

Permuting the $\beta_k$ if necessary, we can assume that $\sum_{k=1}^{n-1}$ is a root (Prop. 19), and hence that $\alpha$ is the sum of the positive roots $\sum_{k=1}^{n-1} \beta_k$ and $\beta_n$.