angles between simple roots are obtuse, problem with proof Let $\Phi$ be a root system in the following sense: (1) $\Phi \subset \mathbb{R}^n$ consists of a finite number of nonzero vectors, (2) for each $\alpha \in \Phi$, $\Phi \cap \mathbb{R} \alpha = \{-\alpha,\alpha\}$ and (3) for each $\alpha \in \Phi$, $s_\alpha \Phi = \Phi$ where $s_\alpha$ is the reflection in $\alpha$.
Let $\Delta \subset \Phi$ be a simple system. Then, for $\alpha \neq \beta$ in $\Delta$, we can show that $(\alpha,\beta) \leq 0$.
Humpfreys (in Reflection Groups and Coxeter Groups) proves this by contradiction, in a weird way. I want to know why my simple argument below does not work (since if it did, he would just do it this way).
Argument: Suppose for a contradiction that $(\beta,\alpha) \geq 0$. Then $s_\alpha \beta = \beta - c\alpha$ where $c = \dfrac{2(\beta,\alpha)}{(\alpha,\alpha)} \geq 0$. Then since $\Delta$ is a basis of $\text{span}_\mathbb{R} \Phi$, every element of $\Phi$ has a unique representation as a linear combination of $\Delta$ with coefficients of all the same sign. This is already a contradiction to $s_\alpha \beta = \beta - c\alpha$.
Where is my misunderstanding?
I would like to emphasise that these root systems are not crystallographic, as one sees with Lie Algebras. 
 A: Your argument is correct. There are two potential misunderstandings. 
The first involves his choice to have this fact appear as a corollary to the main theorem of the section. It is common for authors to choose to apply the main theorem of a section to derive corollaries that really have simple self-contained proofs of their own like the one you gave. He wants to show the power of the section's main theorem, so he uses it to derive the obtuse-angle fact as a corollary.
But I suspect your real confusion is this: The $\Delta$ of the proof of the main theorem is a set he constructs that may or may not be simple, and as such, has to be proven to be simple. Your argument begins with $\Delta$ as a simple system, which is fine for the obtuse-angle fact, but no good for proving the main theorem.
He is trying to prove that every positive root system $\Pi$ in $\Phi$ contains a unique simple system $\Delta$. For the existence portion of the proof, he lets $\Delta \subseteq \Pi$ be a minimal set subject to the requirement that each root in $\Pi$ is a non-negative linear combination of roots in $\Delta$. (The set $\Pi$ is such a set, but is probably not minimal.) He can't assume that $\Delta$ is itself a simple system, because then the argument would be circular. To prove $\Delta$ is a simple system, he has to show that $\Delta$ is a linearly independent set, which he does by proving that for this (not necessarily simple) $\Delta$, the obtuse-angle property holds. Since every simple system is contained in some positive root system $\Pi$, and this constructed $\Delta$ is the unique simple system contained in $\Pi$, it follows that the obtuse-angle property holds for arbitrary simple systems. 
