Elementary Consequences of The Root System Axioms
The cosine of the angle between two roots is constrained to be a half-integral multiple of a square root of an integer. This is because and are both integers, by assumption, and
Since, the only possible values for are, corresponding to angles of 90°, 60° or 120°, 45° or 135°, 30° or 150°, and 0 or 180°. Condition 2 says that no scalar multiples of α other than 1 and -1 can be roots, so 0 or 180°, which would correspond to 2α or -2α are out.
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