Obtuse nonopposite roots have a root sum ($(\alpha ,\beta )<0,\beta \ne -\alpha \Rightarrow \alpha +\beta \in R$)

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.