Figure 5. Generating an algebra's full root system using Weyl reflections
To play the animation, press '<CTRL>-A' to show the animation pane and click 'Start'.

The full set of roots can be obtained from the simple roots using Weyl reflections, as shown in the above animation. Two simple roots, colored red and blue, are drawn according to the rules given by their Dynkin diagram. Additional roots are generated by reflecting the roots across a hyperplane orthogonal to one of the simple roots. This process is repeated until all of the roots have been generated. Notice that only the last root system, that of $G_2$, requires all four Weyl reflections.