  In this paper, we provide a direct and constructive proof of weak
factorization of $h^1(\mathbb{R})$ (the predual of little BMO space
bmo$(\mathbb{R}\times\mathbb{R})$ studied by Cotlar-Sadosky and
Ferguson-Sadosky), i.e., for every $f\in h^1(\mathbb{R}\times\mathbb{R})$ there
exist sequences $\{\alpha_j^k\}\in\ell^1$ and functions $g_j^k,h^k_j\in
L^2(\mathbb{R}^2)$ such that \begin{align*}
f=\sum_{k=1}^\infty\sum_{j=1}^\infty\alpha^k_j\Big(\, h^k_j H_1H_2 g^k_j -
g^k_j H_1H_2 h^k_j\Big) \end{align*} in the sense of $h^1(\mathbb{R})$, where
$H_1$ and $H_2$ are the Hilbert transforms on the first and second variable,
respectively. Moreover, the norm $\|f\|_{h^1(\mathbb{R}\times\mathbb{R})}$ is
given in terms of $\|g^k_j\|_{L^2(\mathbb{R}^2)}$ and
$\|h^k_j\|_{L^2(\mathbb{R}^2)}$. By duality, this directly implies a lower
bound on the norm of the commutator $[b,H_1H_2]$ in terms of $\|b\|_{{\rm
bmo}(\mathbb{R}\times\mathbb{R})}$.
  Our method bypasses the use of analyticity and the Fourier transform, and
hence can be extended to the higher dimension case in an arbitrary
$n$-parameter setting for the Riesz transforms.
