Managing $\gamma_5$ in Dimensional Regularization II: the Trace with more $\gamma_5$

In the present paper we evaluate the anomaly for the abelian axial current in a non abelian chiral gauge theory, by using dimensional regularization. This amount to formulate a procedure for managing traces with more than one $\gamma_5$. \par The suggested procedure obeys Lorentz covariance and cyclicity, at variance with previous approaches (e.g. the celebrated 't Hooft and Veltman's where Lorentz is violated) \par The result of the present paper is a further step forward in the program initiated by a previous work on the traces involving a single $\gamma_5$. The final goal is an unconstrained definition of $\gamma_5$ in dimensional regularization. Here, in the evaluation of the anomaly, we profit of the axial current conservation equation, when radiative corrections are neglected. This kind of tool is not always exploited in field theories with $\gamma_5$, e.g. in the use of dimensional regularization of infrared and collinear divergences.