Let $X$ be an $n$-dimensional smooth manifold, let $m,k\in\mathbb{R}$, and let $P:C_c^\infty(X)\to C^\infty(X)$ and $A:C_c^\infty(X)\to C^\infty(X)$ be properly supported classical scalar Kohn-Nirenberg pseudodifferential operators of orders $m$ and $k$, respectively, acting on scalar functions. Suppose that their principal symbols are represented by smooth scalar functions $p:T^*X\setminus 0\to\mathbb{C}$ and $a:T^*X\setminus 0\to\mathbb{C}$ that are homogeneous of degrees $m$ and $k$ in the cotangent variable, respectively, so that $\sigma_m(P)=p$ and $\sigma_k(A)=a$. Define the commutator by $[P,A]=P\circ A-A\circ P$. In every coordinate chart $(U,\varphi)$ with local coordinates $(x_1,\dots,x_n)$ and induced cotangent fiber coordinates $(\xi_1,\dots,\xi_n)$, use the convention $D_{x_j}=\frac{1}{i}\partial_{x_j}$ and define the Poisson bracket of $p$ and $a$ locally by $\{p,a\}=\sum_{j=1}^n(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{x_j}p\,\partial_{\xi_j}a)$. Then $[P,A]\in \Psi^{m+k-1}(X)$ and $\sigma_{m+k-1}([P,A])=\frac{1}{i}\{p,a\}$.