  Through the calculation of the matrix element of the singlet axial-current
operator between the vacuum and a pair of gluons in dimensional regularization
with an anticommuting $\gamma_5$ defined in a Kreimer-scheme variant, we find
that additional renormalization counter-terms proportional to the Chern-Simons
current operator are needed starting from $\mathcal{O}(\alpha_s^2)$ in QCD.
This is in contrast to the well-known purely multiplicative renormalization of
the singlet axial-current operator defined with a non-anticommuting $\gamma_5$.
Consequently, without introducing compensation terms in the form of additional
renormalization, the Adler-Bell-Jackiw anomaly equation does not hold
automatically in the bare form in this kind of schemes. We determine the
corresponding (gauge-dependent) coefficient to $\mathcal{O}(\alpha_s^3)$ in
QCD, using a variant of the original Kreimer prescription which is implemented
in our computation in terms of the standard cyclic trace together with a
constructively-defined $\gamma_5$. Owing to the factorized form of these
divergences, intimately related to the axial anomaly, we further performed a
check, using concrete examples, that with $\gamma_5$ treated in this way, the
axial-current operator needs no more additional renormalization in dimensional
regularization but \textit{only} for non-anomalous amplitudes in a
perturbatively renormalizable theory. To be complete, we provide a few
additional ingredients needed for a proposed extension of the algorithmic
procedure formulated in the above analysis to potential applications to a
renormalizable anomaly-free chiral gauge theory, i.e.~the electroweak theory.
