  Let $L= -\Delta+ V$ be a Schr\"odinger operator on $\mathbb R^d$, $d\geq 3$,
where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse
H\"older class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for
$T$ in a class $\mathcal K_L$ of sublinear operators containing the fundamental
operators in harmonic analysis related to $L$. More precisely, when $T\in
\mathcal K_L$, we prove that there exists a bounded subbilinear operator
$\mathfrak R= \mathfrak R_T: H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)\to
L^1(\mathbb R^d)$ such that $|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq
|[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|$, where $\mathfrak S$
is a bounded bilinear operator from $H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)$
into $L^1(\mathbb R^d)$ which does not depend on $T$. The subbilinear
decomposition (\ref{abstract 1}) explains why commutators with the fundamental
operators are of weak type $(H^1_L,L^1)$, and when a commutator $[b,T]$ is of
strong type $(H^1_L,L^1)$. Also, we discuss the $H^1_L$-estimates for
commutators of the Riesz transforms associated with the Schr\"odinger operator
$L$.
