  Based on the view that Einstein's theory can be interpreted as a gauge theory
of Lorentz group, we decompose the gravitational connection (the gauge
potential of Lorentz group) $\vGm_\mu$ into the restricted connection made of
the potential of the maximal Abelian subgroup $H$ of Lorentz group $G$ and the
valence connection made of $G/H$ part of the potential which transforms
covariantly under Lorentz gauge transformation. With this decomposition we show
that the Einstein's theory can be decomposed into the restricted part made of
the restricted connection which has the full Lorentz gauge invariance and the
valence part made of the valence connection which plays the role of
gravitational source of the restricted gravity. We show that there are two
different Abelian decomposition of Einstein's theory, the light-like (or null)
decomposition and the non light-like (or non-null) decomposition, because
Lorentz group has two maximal Abelian subgroups. In this decomposition the role
of the metric $g_\mn$ is replaced by a four-index metric tensor $\vg_\mn$ which
transforms covariantly under the Lorentz group, and the metric-compatibility
condition $\nabla_\alpha g_\mn=0$ of the connection is replaced by the gauge
and generally covariant condition ${\mathscr D}_\mu \vg^\mn=0$. The
decomposition shows the existence of a restricted theory of gravitation which
has the full general invariance but is much simpler and has less physical
degrees of freedom than Einstein's theory. Moreover, it tells that the
restricted gravity can be written as an Abelian gauge theory, which implies
that the graviton can be described by a massless spin-one field.
