Let $U \subset \mathbb{R}^n$ be open, let $V \Subset U$ be an open subset, and let $m,m' \in \mathbb{R}$. Let $a \in S^m_{1,0}(U \times \mathbb{R}^n)$ and $b \in S^{m'}_{1,0}(U \times \mathbb{R}^n)$ be symbols, viewed either after localization to $V$ or as properly supported representatives. For each $j \in \{1,\dots,n\}$, let $D_{x_j}$ denote the differential operator $-i\partial_{x_j}$. For a multi-index $\alpha = (\alpha_1,\dots,\alpha_n) \in \mathbb{N}_0^n$, write $D_x^\alpha := D_{x_1}^{\alpha_1}\cdots D_{x_n}^{\alpha_n}$, $\partial_\xi^\alpha := \partial_{\xi_1}^{\alpha_1}\cdots \partial_{\xi_n}^{\alpha_n}$, $|\alpha| := \alpha_1+\cdots+\alpha_n$, and $\alpha! := \alpha_1!\cdots\alpha_n!$.

Let $c \in S^{m+m'}_{1,0}(V \times \mathbb{R}^n)$ be the left-quantized composition symbol, so that on $V$ the left-quantized operator with symbol $c$ represents the composition of the left-quantized operators with symbols $a$ and $b$, modulo smoothing remainders. Assume that $c$ satisfies the left-quantized symbolic composition expansion: for every integer $N \geq 1$,

\begin{align*} c - \sum_{|\alpha|<N}\frac{1}{\alpha!}\partial_\xi^\alpha a\,D_x^\alpha b \in S^{m+m'-N}_{1,0}(V \times \mathbb{R}^n). \end{align*}

Then

\begin{align*} c - ab \in S^{m+m'-1}_{1,0}(V \times \mathbb{R}^n). \end{align*}

Consequently, the principal symbol class of $c$ in $S^{m+m'}_{1,0}(V \times \mathbb{R}^n) / S^{m+m'-1}_{1,0}(V \times \mathbb{R}^n)$ is represented by $ab$.

Moreover,

\begin{align*} c - ab - \sum_{j=1}^n \partial_{\xi_j}a\,D_{x_j}b \in S^{m+m'-2}_{1,0}(V \times \mathbb{R}^n). \end{align*}

Equivalently, modulo $S^{m+m'-2}_{1,0}(V \times \mathbb{R}^n)$,

\begin{align*} c(x,\xi) \equiv a(x,\xi)b(x,\xi)+\sum_{j=1}^n \partial_{\xi_j}a(x,\xi)D_{x_j}b(x,\xi). \end{align*}