File:Integrality of root systems.svg

English: Conditions on the placement of roots in a root system.

Given a root ${\displaystyle \alpha }$ in a root system, this image shows the possible positions for another root ${\displaystyle \beta \neq \pm \alpha }$ within the 2-dimensional subspace that they span.

The blue lines show the possible half-integer values of ${\displaystyle {\frac {(\alpha ,\beta )}{(\alpha ,\alpha )}}}$ and the red circles show the possible half-integer values of ${\displaystyle {\frac {(\alpha ,\beta )}{(\beta ,\beta )}}}$. The black line shows ${\displaystyle (\alpha ,\beta )=0}$ which are all (except 0) possible values for ${\displaystyle \beta }$. The black points show possible values for ${\displaystyle \beta }$ and are labeled with the angle ${\displaystyle {\widehat {\alpha ,\beta }}}$.