I'm planning to teach a course on reflection/Coxeter groups next Fall, and have started outlining the first days, which will presumably be the classification of finite reflection groups. Here is an issue I've run into.
At the end of the classification, one has shown that every finite reflection group is generated by reflections in a set of linearly independent vectors $\alpha_i$ (the simple roots) which I'll normalize to have length $1$, and has described all possible Gram matrices $( \langle \alpha_i, \alpha_j \rangle)$. One then wants to show that all the Gram matrices on the list actually occur.
Given a positive definite Gram matrix $G$, one can always find linearly independent vectors $\alpha_i$ and one can consider the group $W$ generated by reflections in them. But it is not clear that $W$ is finite.
In all but two cases, $W$ is crystallographic, meaning that it preserves a lattice. It is easy to show that a subgroup of $O(n)$ preserving a rank $n$ lattice in $\mathbb{R}^n$ is finite.
The remaining cases are the dihedral groups $I_2^n$, and the groups $H_3$ and $H_4$ (symmetry groups of the icoshaedron and the 600-cell, respectively). I am willing to treat it as obvious that the dihedral groups are finite, and maybe I'm willing to bring in an icosahedron in place of a proof, but I'd rather not just rely on brute force computation to assert that $H_4$ is finite.
For the groups $H_3$ and $H_4$, is there a quick way to see that the group generated by these reflections is finite?
The route which I would most like to take is to prove the following lemma. I know this result is true, and I'll want it later in the course, but I don't know of a proof which can be given before I have built up any tools
Lemma Let $\alpha_1$, ... $\alpha_n$ be unit vectors in $\mathbb{R}^n$ such that each $\langle \alpha_i, \alpha_j \rangle$ is of the form $-\cos \tfrac{\pi}{m_{ij}}$ for $i \neq j$. Let $W$ be the group generated by the reflections in the $\alpha_i$, and let $\Phi = W \cdot \{ \alpha_1, \ldots, \alpha_n \}$. Then every vector in $\Phi$ is either a positive linear combination of the $\alpha_i$ or a negative linear combination of the $\alpha_i$.
If I can prove this, I can establish the result as follows: Take the unit sphere $S^{n-1}$ and cut it along each of the hyperplanes $\beta^{\perp}$ for $\beta \in \Phi$. $W$ acts on the connected components of the sliced sphere. Let $D$ be the region $S^{n-1} \cap \{ \omega : \langle \alpha_i, \omega \rangle > 0 \}$. The hypothesis implies that $D$ is one of the connected components, and it has positive $(n-1)$-dimensional volume. Since $\mathrm{Vol}(S^{n-1}) < \infty$, the $W$-orbit of $D$ must be finite, and it is easy to show that the stabilizer of $D$ is trivial. But I don't know if I can prove this lemma with no tools.
