Please regard this A COMBINATORIAL CONSTRUCTION FOR SIMPLY–LACED LIE ALGEBRAS on page 7 (it is brief but I hope that page is enought introduction on that topic).
Can I argue, and if so how, that the coxeter group $W$ is finite if the set $R(X)$ is finite (i.e. first $\Longrightarrow$ in Proposition 3.2)? Somewhere I read that a Coxeter group does not neccessarily have to be finite even if the generator set is. Lemma 3.1 states that my $m_{xy}$ (see Coxeter group definition) are 2 or 3 depending on the relation between $x\neq y$ which I already checked and is true.
EDIT: Ok page with paper seems to be down. Updated link.