Let $n \in \mathbb{N}$, let $m,m' \in \mathbb{R}$, and let $h_0 > 0$. For each $\ell \in \mathbb{R}$, let $S^\ell(T^*\mathbb{R}^n)$ denote the semiclassical symbol class of families $p=(p_h)_{0<h\leq h_0}$ with $p_h \in C^\infty(T^*\mathbb{R}^n;\mathbb{C})$ whose defining symbol seminorm estimates are uniform for $h \in (0,h_0]$. Let $a=(a_h)_{0<h\leq h_0} \in S^m(T^*\mathbb{R}^n)$ and $b=(b_h)_{0<h\leq h_0} \in S^{m'}(T^*\mathbb{R}^n)$. For each $h \in (0,h_0]$, let $\operatorname{Op}_h^w(a_h)$ and $\operatorname{Op}_h^w(b_h)$ denote the semiclassical Weyl quantizations acting continuously from $\mathcal{S}(\mathbb{R}^n)$ to $\mathcal{S}'(\mathbb{R}^n)$. Define the Poisson bracket $\{a,b\}=(\{a_h,b_h\})_{0<h\leq h_0}: T^*\mathbb{R}^n \to \mathbb{C}$ by

\begin{align*} \{a_h,b_h\}(x,\xi) = \sum_{j=1}^n \partial_{\xi_j}a_h(x,\xi)\,\partial_{x_j}b_h(x,\xi) - \partial_{x_j}a_h(x,\xi)\,\partial_{\xi_j}b_h(x,\xi). \end{align*}

Then there exists a symbol family $r_3=(r_{3,h})_{0<h\leq h_0} \in S^{m+m'-3}(T^*\mathbb{R}^n)$, with symbol seminorms uniform for $h \in (0,h_0]$, such that, modulo residual semiclassical operators in $\Psi_h^{-\infty}(\mathbb{R}^n)$,

\begin{align*} [\operatorname{Op}_h^w(a_h),\operatorname{Op}_h^w(b_h)] = \frac{h}{i}\operatorname{Op}_h^w(\{a_h,b_h\}) + h^3\operatorname{Op}_h^w(r_{3,h}). \end{align*}

Here $\Psi_h^\ell(\mathbb{R}^n)$ denotes the class of semiclassical pseudodifferential operators of order $\ell$ on $\mathbb{R}^n$, and $\Psi_h^{-\infty}(\mathbb{R}^n)$ denotes the residual class. For any fixed non-Weyl semiclassical quantization $\operatorname{Op}_h^\kappa:S^\ell(T^*\mathbb{R}^n)\to \Psi_h^\ell(\mathbb{R}^n)$ whose symbolic composition formula has first antisymmetric coefficient equal to the Poisson bracket, the same principal commutator term occurs, while the remainder is generally $h^2R_h$ for some $R_h \in \Psi_h^{m+m'-2}(\mathbb{R}^n)$, unless an additional symmetry cancels the second-order term.