The dihedral angle between two of the root hyperplanes in a root system can be , , or degrees and nothing else.
Statement of result
By the angle between two hyperplanes, I mean the following:
take the unit vectors and orthogonal to these hyperplanes,
take the smallest angle such that .
In a root system, the vectors orthogonal to the root hyperplanes are the (simple) roots, so we are really interested in where and are roots. Recall that the dot product is the Killing form.
Since we're in a root system, if and are roots then for some , where is the orthogonal projection of onto the line through .
In the root system shown below, the projection of to the line through gives .
By Euclidean geometry, we have
This means that We also have so
where is the angle between the roots.
We have because the roots are not collinear, so because it's an integer between 0 and 4.
These possibilities give degrees.