On central sequence algebras of tensor product von Neumann algebras

We show that when $M,N_1,N_2$ are tracial von Neumann algebras with $M^{\mathrm{\prime }}\cap M^{\omega }$ abelian, $M^{\mathrm{\prime }}\cap (M\bar{\otimes }{N}_1{)}^{\omega }$ and $M^{\mathrm{\prime }}\cap (M\bar{\otimes }{N}_2{)}^{\omega }$ commute in $(M\bar{\otimes }{N}_1\bar{\otimes }{N}_2{)}^{\omega }$. As a consequence, we obtain information on McDuff decompositions of ${\mathrm{I}\mathrm{I}}_1$ factors of the form $M\bar{\otimes }N$, where $M$ is a non-McDuff factor.