[proofplan]
We compute in a coordinate chart using the standard local composition formula for properly supported classical pseudodifferential operators. The leading order $m+k$ terms in the complete symbols of $PA$ and $AP$ are $pa$ and $ap$, which cancel because the principal symbols are scalar. The next order term comes only from the $|\alpha|=1$ part of the composition formula, and with the convention $D_{x_j}=\frac{1}{i}\partial_{x_j}$ this term is exactly $\frac{1}{i}\{p,a\}$. Since this expression is the Poisson bracket on $T^*X\setminus 0$, it transforms invariantly under coordinate changes and therefore defines the global principal symbol.
[/proofplan]
[step:Choose local complete symbols and apply the composition formula]
Work in a coordinate chart $(U,\varphi)$ on $X$, with induced local coordinates $(x_1,\dots,x_n)$ on $U$ and fiber coordinates $(\xi_1,\dots,\xi_n)$ on $T^*U$. Let
\begin{align*}
q_P:T^*U\setminus 0\to \mathbb{C}
\end{align*}
be a local classical Kohn-Nirenberg complete symbol for $P$ in this chart, understood modulo smoothing symbols, with homogeneous expansion
\begin{align*}
q_P\sim \sum_{\ell=0}^{\infty} p_{m-\ell},
\end{align*}
where $p_m=p$ and each $p_{m-\ell}$ is homogeneous of degree $m-\ell$ in $\xi$. Similarly, let
\begin{align*}
q_A:T^*U\setminus 0\to \mathbb{C}
\end{align*}
be a local classical Kohn-Nirenberg complete symbol for $A$, understood modulo smoothing symbols, with homogeneous expansion
\begin{align*}
q_A\sim \sum_{\ell=0}^{\infty} a_{k-\ell},
\end{align*}
where $a_k=a$ and each $a_{k-\ell}$ is homogeneous of degree $k-\ell$ in $\xi$.
By the standard local composition theorem for properly supported Kohn-Nirenberg pseudodifferential operators, applied in this coordinate chart with the convention $D_{x_j}=\frac{1}{i}\partial_{x_j}$, the complete symbols of $PA$ and $AP$ have asymptotic expansions
\begin{align*}
q_{PA}\sim \sum_{\alpha\in \mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha q_P\,D_x^\alpha q_A
\end{align*}
and
\begin{align*}
q_{AP}\sim \sum_{\alpha\in \mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha q_A\,D_x^\alpha q_P.
\end{align*}
Here $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index,
\begin{align*}
\partial_\xi^\alpha=\partial_{\xi_1}^{\alpha_1}\cdots \partial_{\xi_n}^{\alpha_n}
\end{align*}
and
\begin{align*}
D_x^\alpha=D_{x_1}^{\alpha_1}\cdots D_{x_n}^{\alpha_n}.
\end{align*}
Thus $PA$ and $AP$ are pseudodifferential operators of order $m+k$, and their difference $[P,A]$ has local complete symbol $q_{PA}-q_{AP}$.
[guided]
We first reduce the statement to a local symbol calculation. Choose a coordinate chart $(U,\varphi)$ on $X$, write $(x_1,\dots,x_n)$ for the induced coordinates on $U$, and write $(\xi_1,\dots,\xi_n)$ for the corresponding cotangent fiber coordinates on $T^*U$. In this chart, choose a local classical Kohn-Nirenberg complete symbol
\begin{align*}
q_P:T^*U\setminus 0\to \mathbb{C}
\end{align*}
for $P$, understood modulo smoothing symbols. Since $P$ has order $m$ and principal symbol represented by $p$, its homogeneous expansion begins as
\begin{align*}
q_P\sim p+p_{m-1}+p_{m-2}+\cdots,
\end{align*}
where $p$ is homogeneous of degree $m$ in the covariable $\xi$. Likewise, choose a local classical Kohn-Nirenberg complete symbol
\begin{align*}
q_A:T^*U\setminus 0\to \mathbb{C}
\end{align*}
for $A$, understood modulo smoothing symbols, with expansion
\begin{align*}
q_A\sim a+a_{k-1}+a_{k-2}+\cdots,
\end{align*}
where $a$ is homogeneous of degree $k$ in $\xi$.
The main tool is the standard local composition theorem for properly supported Kohn-Nirenberg pseudodifferential operators. The proper support hypothesis ensures that the compositions $PA$ and $AP$ are again defined as pseudodifferential operators, and the classical symbol hypothesis ensures that the asymptotic expansions may be compared degree by degree. With the convention $D_{x_j}=\frac{1}{i}\partial_{x_j}$, the composition formula gives
\begin{align*}
q_{PA}\sim \sum_{\alpha\in \mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha q_P\,D_x^\alpha q_A
\end{align*}
and
\begin{align*}
q_{AP}\sim \sum_{\alpha\in \mathbb{N}_0^n}\frac{1}{\alpha!}\partial_\xi^\alpha q_A\,D_x^\alpha q_P.
\end{align*}
Here $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index,
\begin{align*}
\partial_\xi^\alpha=\partial_{\xi_1}^{\alpha_1}\cdots \partial_{\xi_n}^{\alpha_n}
\end{align*}
and
\begin{align*}
D_x^\alpha=D_{x_1}^{\alpha_1}\cdots D_{x_n}^{\alpha_n}.
\end{align*}
This is the point where all analytic input enters the proof: after the composition formula is available, the rest is bookkeeping by homogeneous degree. The commutator $[P,A]=PA-AP$ has complete symbol $q_{PA}-q_{AP}$ in the same chart, so we now compare the two displayed expansions.
[/guided]
[/step]
[step:Cancel the order $m+k$ part using scalarity of the principal symbols]
The order $m+k$ part of $q_{PA}$ comes only from the $\alpha=0$ term using the leading homogeneous pieces of $q_P$ and $q_A$, hence it is $pa$. The order $m+k$ part of $q_{AP}$ is similarly $ap$. Because $p$ and $a$ are scalar-valued functions, $pa=ap$. Therefore the homogeneous component of degree $m+k$ in $q_{PA}-q_{AP}$ vanishes, so
\begin{align*}
[P,A]\in \Psi^{m+k-1}(U).
\end{align*}
Since the argument is local and pseudodifferential order is local, this gives
\begin{align*}
[P,A]\in \Psi^{m+k-1}(X).
\end{align*}
[/step]
[step:Extract the homogeneous term of degree $m+k-1$]
We now compute the degree $m+k-1$ part of $q_{PA}-q_{AP}$. Contributions from $\alpha=0$ involving $p_{m-1}$ or $a_{k-1}$ are
\begin{align*}
p\,a_{k-1}+p_{m-1}\,a
\end{align*}
for $PA$, and
\begin{align*}
a\,p_{m-1}+a_{k-1}\,p
\end{align*}
for $AP$. These cancel because all symbols act on scalar functions.
For $|\alpha|=1$, write $e_j\in \mathbb{N}_0^n$ for the multi-index with $1$ in the $j$th entry and $0$ elsewhere. The degree $m+k-1$ contribution to $q_{PA}$ is
\begin{align*}
\sum_{j=1}^n \partial_{\xi_j}p\,D_{x_j}a,
\end{align*}
and the corresponding contribution to $q_{AP}$ is
\begin{align*}
\sum_{j=1}^n \partial_{\xi_j}a\,D_{x_j}p.
\end{align*}
Terms with $|\alpha|\geq 2$ have degree at most $m+k-2$, because each $\xi$-derivative lowers the homogeneous degree by one while each $x$-derivative preserves it. Hence the homogeneous component of degree $m+k-1$ in the commutator symbol is
\begin{align*}
\sum_{j=1}^n\left(\partial_{\xi_j}p\,D_{x_j}a-\partial_{\xi_j}a\,D_{x_j}p\right).
\end{align*}
Using $D_{x_j}=\frac{1}{i}\partial_{x_j}$, this becomes
\begin{align*}
\frac{1}{i}\sum_{j=1}^n\left(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{\xi_j}a\,\partial_{x_j}p\right).
\end{align*}
Since scalar multiplication commutes, the second term may be rewritten as
\begin{align*}
\partial_{\xi_j}a\,\partial_{x_j}p=\partial_{x_j}p\,\partial_{\xi_j}a.
\end{align*}
Therefore the degree $m+k-1$ term is
\begin{align*}
\frac{1}{i}\sum_{j=1}^n\left(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{x_j}p\,\partial_{\xi_j}a\right)=\frac{1}{i}\{p,a\}.
\end{align*}
[guided]
The next task is to identify exactly which terms can still contribute after the order $m+k$ cancellation. There are two possible sources of degree $m+k-1$: the $\alpha=0$ part using one subprincipal homogeneous term, and the $|\alpha|=1$ part using the two leading homogeneous terms.
First consider $\alpha=0$. In $q_{PA}$, the degree $m+k-1$ part coming from $\alpha=0$ is
\begin{align*}
p\,a_{k-1}+p_{m-1}\,a.
\end{align*}
In $q_{AP}$, the corresponding expression is
\begin{align*}
a\,p_{m-1}+a_{k-1}\,p.
\end{align*}
These expressions cancel in the difference because the symbols are scalar-valued functions:
\begin{align*}
p\,a_{k-1}=a_{k-1}\,p
\end{align*}
and
\begin{align*}
p_{m-1}\,a=a\,p_{m-1}.
\end{align*}
This cancellation is precisely why the theorem assumes scalar principal symbols; for matrix-valued leading symbols, an order $m+k$ matrix commutator can remain.
Now consider the terms with $|\alpha|=1$. For each $j\in\{1,\dots,n\}$, let $e_j\in\mathbb{N}_0^n$ denote the multi-index with $1$ in the $j$th position and $0$ elsewhere. The $e_j$ term in the composition formula for $PA$ is
\begin{align*}
\partial_{\xi_j}q_P\,D_{x_j}q_A.
\end{align*}
Taking the leading homogeneous part of this expression gives
\begin{align*}
\partial_{\xi_j}p\,D_{x_j}a,
\end{align*}
because $\partial_{\xi_j}p$ has degree $m-1$ in $\xi$ and $D_{x_j}a$ has degree $k$ in $\xi$. Thus the degree is $m+k-1$. Summing over $j$ gives the degree $m+k-1$ contribution
\begin{align*}
\sum_{j=1}^n \partial_{\xi_j}p\,D_{x_j}a.
\end{align*}
The same argument applied to $AP$ gives
\begin{align*}
\sum_{j=1}^n \partial_{\xi_j}a\,D_{x_j}p.
\end{align*}
No term with $|\alpha|\geq 2$ can contribute to degree $m+k-1$. Each derivative $\partial_{\xi_j}$ lowers the homogeneous degree by one, while $D_{x_j}$ differentiates only in the base variable and does not change the homogeneous degree in $\xi$. Therefore a term with $|\alpha|\geq 2$ has degree at most $m+k-2$ when applied to the two leading homogeneous symbols, and lower-order homogeneous pieces only decrease the degree further.
Consequently the homogeneous component of degree $m+k-1$ in the commutator symbol is
\begin{align*}
\sum_{j=1}^n\left(\partial_{\xi_j}p\,D_{x_j}a-\partial_{\xi_j}a\,D_{x_j}p\right).
\end{align*}
Now insert the convention $D_{x_j}=\frac{1}{i}\partial_{x_j}$. This gives
\begin{align*}
\frac{1}{i}\sum_{j=1}^n\left(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{\xi_j}a\,\partial_{x_j}p\right).
\end{align*}
Since the symbols are scalar-valued, multiplication commutes, so
\begin{align*}
\partial_{\xi_j}a\,\partial_{x_j}p=\partial_{x_j}p\,\partial_{\xi_j}a.
\end{align*}
Thus the expression becomes
\begin{align*}
\frac{1}{i}\sum_{j=1}^n\left(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{x_j}p\,\partial_{\xi_j}a\right).
\end{align*}
By the stated coordinate formula for the Poisson bracket on $T^*U$, this is exactly
\begin{align*}
\frac{1}{i}\{p,a\}.
\end{align*}
[/guided]
[/step]
[step:Patch the local formula to a global principal symbol]
The local expression
\begin{align*}
\sum_{j=1}^n\left(\partial_{\xi_j}p\,\partial_{x_j}a-\partial_{x_j}p\,\partial_{\xi_j}a\right)
\end{align*}
is the coordinate formula for the Poisson bracket determined by the canonical symplectic form on $T^*X$. Hence the degree $m+k-1$ homogeneous term computed above is invariant under coordinate changes. The local principal symbols therefore agree on overlaps of coordinate charts and patch to the global principal symbol
\begin{align*}
\sigma_{m+k-1}([P,A])=\frac{1}{i}\{p,a\}
\end{align*}
on $T^*X\setminus 0$. This proves both the asserted order drop and the principal symbol formula.
[/step]
