Let $(\Delta,(-,-))$ be a root system and $\alpha,\beta \in \Delta$ be s.t. $(\alpha,\beta) = 0$. Is it always true that $$ \alpha + \beta, ~ \alpha - \beta \notin \Delta? $$ This works for the $A$-series, but is there a general argument?
2026-03-26 14:42:10.1774536130
The sum of two roots when their pairing is zero
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In a root system $R$ of type $B_2$, there are various pairs of short roots $\alpha, \beta$ with $(\alpha,\beta)=0$ but both $\alpha \pm \beta \in R$.