Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an element in its root system $\Delta$ is the set of the roots of the same length?
I have shown this for the ADE types of groups, which is quite easy since if G is an ADE group, then for any element $r_i, r_j\in\Delta, \exists T\in G: Tr_i=r_j$. Thus the orbit of any single element in the root system of a group of the ADE type is just the entire root system itself, which are of course roots of the same length because ADE groups have roots of the same length.
But how would I show this for the non ADE types? ie. $F_4, G_2, B_n$