  We exhibit a new consistent group-manifold reduction of pure Einstein gravity
in the vielbein formulation when the compactification group manifold is S^3.
The novel feature in the reduction is to exploit the two 3-dimensional Lie
algebras that S^3 admits. The first algebra is introduced into the
group-manifold reduction in the standard way through the Maurer-Cartan 1-forms
associated to the symmetry of the general coordinate transformations. The
second algebra is associated to the linear adjoint group and it is introduced
into the group-manifold reduction through a local transformation in the
internal tangent space. We discuss the characteristics of the resulting
lower-dimensional theory and we emphasize the novel results generated by the
new group-manifold reduction. As an application of the reduction we show that
the lower-dimensional theory admits a domain wall solution which upon uplifting
to the higher-dimension results to be the self-dual (in the non-vanishing
components of both curvature and spin connection) Kaluza-Klein monopole. This
discussion may be relevant in the dimensional reductions of M-theory, string
theory and also in the Bianchi cosmologies in four dimensions.
