Integrality over fixed rings of automorphisms in a Lie nilpotent setting

Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].