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

Supertranslation invariance of angular momentum at null infinity in double null gauge