  Under the assumption that the color charge can be written in a BRST exact
form, the color confinement mechanism proposed by Kugo and Ojima (KO) explains
the confinement of any colored particles including dynamical quarks and gluons.
This mechanism, however, is known to break down in the Abelian gauge which
treats the maximal Abelian subgroup of the gauge group in a special manner. In
order to study whether the failure of the KO mechanism is particular only to
the Abelian gauge or whether this failure occurs in a wide class of gauges
including the ordinary Lorentz type gauge, we carry out a renormalization group
study of the $SU(2)$ gauge theory in the gauge fixing space. Our gauge fixing
space consists of four distinct regions that are not connected with each other
by renormalization group flows, and we find that the Abelian gauge is {\it
infrared unstable} in three regions which include the Lorentz type gauge. This
suggests that the failure of the KO mechanism is a phenomenon which occurs only
in the Abelian gauge. We also find that the Lorentz gauge is infrared stable.
