I am trying to calculate the angles of the fundamental alcove (chamber?) for the root systems of type $B_2$, $C_2$, and $G_2$; the fundamental alcove (chamber?) forms a triangle in these cases, so I am trying to calculate the angles in a triangle.
Given a root system $\Phi$ with simple system $\Delta$,
$$H_{\alpha} = \{\lambda \in V : \langle \lambda, \alpha \rangle = 0 \}$$ are the hyperplanes and the connected components of
$$V \setminus \bigcup_{\alpha} H_{\alpha}$$ are the chambers. Given a chamber $C$ corresponding to a simple system $\Delta$, the walls are defined to be the hyperplanes $H_{\alpha}$ for $\alpha \in \Delta$.
So, from what I gather, to find the angles of the fundamental alcove/chamber I need to compute the angle between these hyperplanes.
So, for the $B_2$ case, I believe the simple roots are $e_1 - e_2$ and $e_2$. However, how can there be a triangle formed by two vectors? Even ignoring this, the angle between these two vectors doesn't come out to the right answer. For the $B_2$ case, the answer is supposed to be $\pi/2, \pi/4, \pi/4$. And, unless I made a mistake in my calculation, the angle between $e_1-e_2$ and $e_2$ is not any of these angles.
What am I doing wrong? What am I misunderstanding?