According to the following wikipedia page https://en.wikipedia.org/wiki/Root_system,
given an irreducible root system, there can be at most two values for the length of the roots.
Because of this, the terms long and short roots make sense. Since the root systems are classified, we can check this case by case. But this does not seem to be a good way to prove the above statement.
Is there a more fundamental explanation for this phenomenon? If so, where can I find such proof?