  By extending the approximate weak factorization method to the bilinear
setting we identify testing conditions on the function $b$ that characterize
the $L^p\times L^q\to L^r$ boundedness of the commutator $[b,T]_1(f,g) =
bT(f,g) - T(bf,g),$ for many exponents in the range $1<p,q<\infty$ and
$r>\frac{1}{2},$ where $T$ is a bilinear Calder\'on-Zygmund operator.
