  The supertranslation invariance of the Chen-Wang-Yau (CWY) angular momentum
in the Bondi-Sachs formalism/gauge was ascertained by the authors in
\cite{CKWWY_evol, CWWY_atmp}. In this article, we study the corresponding
problem in the double null gauge. In particular, supertranslation ambiguity of
this gauge is identified and the CWY angular momentum is proven to be free of
this ambiguity. A similar result is obtained for the CWY center of mass
integral.
