  Certain Iwahori double cosets in the loop group of a reductive group, known
under the names of $P$-alcoves or $(J,w,\delta)$-alcoves, play an important
role in the study of affine Deligne-Lusztig varieties. For such an Iwahori
double coset, its Newton stratification is related to the Newton stratification
of an Iwahori double coset in a Levi subgroup. We show that this relation is
closer than previously known.
