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 $\alpha$.
Questions:
What can we say about the angles between the roots? By looking at the pictures of root systems, it seems that two adjacent roots have an angle of $2\pi/n$ between them... how can this be stated and shown rigorously?
If nothing can be said for question 1 (which I doubt), then can one say something about the angles between simple roots?
This isn't really in the context of Lie algebras, so much as reflection and coxeter groups (I'm studying Humpfrey's Reflection Groups and Coxeter Groups).
Wlog. $\Phi$ spans $\mathbb R^n$. If the angle between two roots $\alpha_1,\alpha_2$ is $\beta$, then $s_{\alpha_1}\circ s_{\alpha_2}$ is a rotation by $2\beta$. Viewed as a permutation of $\Phi$, this rotation has some finite order $k$. As $\Phi$ spand the whole space, the rotation itself also has order $k$, i.e. $2k\beta$ is a multiple of $2\pi$.
For simple roots, we can say even more: The angle between these is one of $\frac12\pi$, $\frac23\pi$, $\frac34\pi$, $\frac56\pi$ (in other words: it is $\pi-\frac1k\pi$ with $k\in\{1,2,3,4,6\}$). One shows this by checking which values the scalar product can assume.