  This paper complements a new lattice formulation of SU(2) Yang-Mills theory
written in terms of new variables in a compact form proposed in the previous
paper. The new variables used in the formulation were once called the
Cho--Faddeev--Niemi or Cho--Faddeev--Niemi--Shabanov decomposition. Our
formulation enables us to explain the infrared ``Abelian'' dominance, in
addition to magnetic monopole dominance shown in the previous paper, in the
gauge invariant way without relying on the specific gauge fixing called the
maximal Abelian gauge used in the conventional investigations. In this paper,
especially, we demonstrate by numerical simulations that gluon degrees of
freedom other than the ``Abelian'' part acquire the mass to be decoupled in the
low-energy region leading to the infrared Abelian dominance.
