TITLE: why most of the angles are right
QUESTION [16 upvotes]: The Coxeter–Dynkin diagrams tell us that in a spherical Coxeter simplex most of the dihedral angles are right. Say among $\tfrac{n{\cdot}(n+1)}2$ dihedral angles we can have at most $n$ angles which are not right.

Is it possible to see this statement without classification?

REPLY [12 votes]: Let the root system be $v_1$, …, $v_n$ with all elements normalized to be length $1$. So $\langle v_i, v_i \rangle =1$, we have $\langle v_i, v_j \rangle \leq - \cos (\pi/3) = -1/2$ for at least $n$ pairs $(i,j)$, and we have $\langle v_i, v_j \rangle \leq 0$ for all $i \neq j$.
But then
$$\left\langle \sum_{i=1}^n v_i, \sum_{i=1}^n v_i \right\rangle \leq  n + 2 n (-1/2) =0,$$
a contradiction.