  For a bounded singular integral $T_n$ in $\mathbb{R}^n$ and a bounded
singular integral $T_m$ in $\mathbb{R}^m$ we prove that $$ \| [T_n^1, [b,
T_m^2]] \|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}}
\|b\|_{\operatorname{BMO}_{\textrm{prod}}(\nu)}, $$ where $p \in (1,\infty)$,
$\mu, \lambda \in A_p$ and $\nu := \mu^{1/p}\lambda^{-1/p}$. Here $T_n^1$ is
$T_n$ acting on the first variable, $T_m^2$ is $T_m$ acting on the second
variable, $A_p$ stands for the bi-parameter weights of $\mathbb{R}^n \times
\mathbb{R}^m$ and $\operatorname{BMO}_{\textrm{prod}}(\nu)$ is a weighted
product BMO space.
