A geometrical construction for the polynomial invariants of some reflection groups

In these notes we investigate the rings of real polynomials in four variables, which are invariant under the action of the reflectiongroups [3,4,3] and [3,3,5]. It is well known that they are rationally generated in degree 2,6,8,12 and 2,12,20,30. We give a different proof of this fact by giving explicit equations for the generating polynomials.