$[b,T] = bT - T(b\,\cdot)$. Already in [@CRW] it was shown that $b\in{\operatorname{BMO}}$ is a sufficient condition for the boundedness of the iterated commutator $[b,[b,\dots,[b,T]]]$ and the same argument extends to the case of commutators $[b_k,[b_{k-1},\dots,[b_1,T]]]$ with different functions, all in ${\operatorname{BMO}}$ separately.

Our object is to make the first systematic study of the iterated commutator $[b_2,[b_1,T]]$ in the case of two different functions $b_1,b_2.$ In particular, we want to identify a joint condition on the pair $(b_1, b_2)$ that is weaker than the individual conditions $b_1, b_2 \in {\operatorname{BMO}}$, that is as close to optimal as possible, and which still guarantees the boundedness of the commutator. This is, in some sense, similar in spirit to the case of bilinear weighted theory, where $w_1, w_2 \in A_4$ is not the optimal condition for the boundedness of bilinear singular integrals from $L^4(w_1) \times L^4(w_2)$ to $L^2(w_1^{1/2}w_2^{1/2})$, but rather there is a genuinely bilinear joint condition $(w_1, w_2) \in A_{(4,4)}$ introduced by Lerner, Ombrosi, Pérez, Torres and Trujillo-Gonzáez [@LOPTT]. In the weighted case the identification of this genuinely bilinear condition has been highly impactful.

We study two-sided estimates for the $L^2 \to L^2$ norm of the commutator $[b_2,[b_1,T]]$. While the upper bounds 