  In order to check the compatibility of the gluon Reggeization in QCD with the
$s$-channel unitarity, the one-loop correction to the Reggeon-Reggeon-gluon
vertex must be known at arbitrary space-time dimension $D$. We obtain this
correction from the gluon production amplitude in the multi-Regge kinematics
and present an explicit expression for it in terms of a few integrals over the
transverse momenta of virtual particles. The one-gluon contribution to the
non-forward BFKL kernel at arbitrary $D$ is also obtained.
