Elements with finite Coxeter part in an affine Weyl group

Let Wa be an affine Weyl group and η:Wa→W0 be the natural projection to the corresponding finite Weyl group. We say that w∈Wa has finite Coxeter part if η(w) is conjugate to a Coxeter element of W0. The elements with finite Coxeter part are a union of conjugacy classes of Wa. We show that for each conjugacy class O of Wa with finite Coxeter part there exists a unique maximal proper parabolic subgroup WJ of Wa, such that the set of minimal length elements in O is exactly the set of Coxeter elements in WJ. Similar results hold for twisted conjugacy classes.