Corollary

Let $\alpha$ and $\beta$ be two roots.

(i) If $(\alpha|\beta) > 0$, $\alpha - \beta$ is a root unless $\alpha = \beta$. (ii) If $(\alpha|\beta) < 0$, $\alpha + \beta$ is a root unless $\alpha = -\beta$. (iii) If $\alpha - \beta \notin \mathbf{R} \cup \{0\}$ and $\alpha + \beta \notin \mathbf{R} \cup \{0\}$, then $(\alpha|\beta) = 0$.

Assertions (i) and (ii) follow from Th. 1 and formula (7) of no. 1. Assertion (iii) follows from (i) and (ii).

It is possible that $\alpha + \beta \in \mathbf{R}$, $(\alpha|\beta) = 0$ (cf. Plate X, System B_2). When $\alpha - \beta \notin \mathbf{R} \cup \{0\}$ and $\alpha + \beta \notin \mathbf{R} \cup \{0\}$, $\alpha$ and $\beta$ are said to be strongly orthogonal.