I have a question regarding a remark in the book "Reflection Groups And Coxeter Groups" by James E. Humphreys (unfortunately the book is not to be found as a whole on google books or such).
In chapter 2 we encounter the definition of a crystallographic root system ($\varPhi$ crystallographic iff $\frac{2(a,b)}{(b,b)} \in \mathbb{Z}\forall a,b \in \varPhi$) and its associated Weyl group $W$ (the group generated by all reflections belonging to elements of $\varPhi$).
After the definition Humphreys follows up with some facts about crystallographic root systems. It is quite easy to see that in the case where $\varPhi$ is irreducible (i.e. the Coxeter graph is connected) there can occur at most two squared length of roots (the roots are then called long and short roots respectively) and $W$ acts transitively on the long and the short roots. After fixing a simple system $\Delta \subset \varPhi$ a partial ordering is introduced on $\varPhi$ via $a \leq b$ iff $b-a$ is a non-negative linear combination of $\Delta$ and the author writes that there is a unique greatest element of $\varPhi$ with respect to this ordering which is a long root.
I can show (using transitivity and a fundamental domain given earlier in the book) that there can be at most two greatest elements one being a long root and one a short root. However I am unable to see why the long root should be greater than the short root. Is there a proof for this fact (preferably one that does not use Lie theory) that does not use the classification and construction of all possible root systems (this is at the moment the only way I could find)?
Any help would be appreciated. Thanks in advance.
Edit: As requested the definition of a simple system: A subset $\Delta \subset \varPhi$ of a rootsystem $\varPhi$ is called simple if $\Delta$ is a basis for the span of $\varPhi$ which satisfies the additional condition that any element of $\varPhi$ is either a non-negative or a non-positive linear combination of $\Delta$.