In the previous paper, we gave a characterization of backward shift invariant subspaces of the Hardy space over the bidisk on which [${S_z}^n$, $S_w^*$] = 0 for a positive integer n ${geq}$ 2. In this case, it holds that ${S_z}^n=cI$ for some $c{in}mathbb{C}$. In this paper, it is proved that if [$S_{varphi}$, $S_w^*$] = 0 and ${varphi}{in}H^{infty}({Gamma}_z)$, then $S_{varphi}=cI$ for some $c{in}mathbb{C}$.