= Obtuse nonopposite roots have a root sum
{title2=$(\alpha,\beta)<0,\ \beta\ne-\alpha\Rightarrow\alpha+\beta\in R$}
For nonproportional roots in a <reduced root system> which is a <crystallographic root system>, the negative <Cartan integers> $m=2(\alpha,\beta)/(\alpha,\alpha)$ and $n=2(\alpha,\beta)/(\beta,\beta)$ satisfy $mn=4\cos^2\theta\in\{1,2,3\}$. At least one of $m,n$ equals $-1$, so reflecting one root in the other produces their sum. The exclusion of opposite roots is essential: their inner product is negative but their sum is zero, which is not a <root>.
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