I'll parallel your book's proof in my language of Lemma 3 in our booklet, but now for the off-diagonal Wigner function, setting $\hbar=1$ , easy to reinstate by dimensional analysis.
The time dependence is a canard, taken up at the very end: the crucial piece, including Wolfgang's integration by parts, visible further below, is for the stationary Wigner function,
$$
W_{E~E'}(x,p)=\frac{1}{2\pi} \int dy e^{-iyp} \tilde f(x,y),
$$
where the all-central cross-spectral density amounts to
$$
\tilde f(x,y)=\langle x+y/2| (|E' \rangle\langle E|)|x-y/2 \rangle \\ =\psi^*_E (x-y/2)~~\psi_{E'} (x+y/2),
$$
for TISE,
$$
\left (-\frac{1}{2m} \partial_x^2 + V(x)\right ) \psi_E (x) = E \psi_E (x).
$$
Thus, by virtue of our eqn (12), the celebrated Bopp shift,
\begin{eqnarray}
&& ~~~~H(x,p)\star W_{EE'}(x,p)= \nonumber \\
&=& {1\over 2\pi} \left( \frac{1}{2m}\left( p-{i\over 2} \overrightarrow{\partial }_x \right) ^2
+V(x) \right) \int\! dy~ e^{-iy(p+{i\over2} \overleftarrow {\partial }_x)}
\tilde f(x,y) \nonumber \\
&=& {1\over 2\pi} \int\! dy~ e^{-iyp} ~ \left(\frac{1}{2m} \left( p-{i\over2} \overrightarrow{\partial }_x\right) ^2
+V(x+y/2)\right) \tilde f(x,y) \nonumber \\
&=& {1\over 2\pi} \int\! dy ~e^{-iyp}
\left( \frac{1}{2m}\left( i\overrightarrow{\partial }_y +{i\over2} \overrightarrow{\partial }_x\right) ^2
+ V(x+ y/2)\right)
\psi^*_E(x- y/2) ~\psi_{E'}(x+ y/2) \nonumber
\\ &=& {1\over 2\pi} \int\! dy ~e^{-iyp} \psi^*(x- y/2)~
E'~ \psi(x+ y/2) = E' ~W_{EE'}(x,p)~.
\end{eqnarray}
The Fourier integration by parts yields the penultimate line, as the surface terms vanish: all wavefunctions, potentials, etc, are assumed to vanish at infinity. In the penultimate line, the operators only Schroedinger-act on the $\psi$, leaving the $\psi^*$ alone! (Think of left- versus right-movers.)
Mutatis mutandis,
$$
W_{EE'}(x,p) \star H (x,p)= E W_{EE'}(x,p).
$$
So, then
$$
W_{EE'} \star H (x,p) + H\star W_{EE'}= (E+E') W_{EE'} ,
$$
and
$$
W_{EE'} \star H (x,p) - H\star W_{EE'}= (E-E') W_{EE'} ,
$$
for these stationary Wigner functions. Consequently, for the time-dependent ones,
$$
{\cal W}_{EE'}(x,p;t) = e^{i(E-E')t} W_{EE'}(x,p),
$$
one gets Moyal's evolution equation for the Wigner function, the Wigner transform of the von Neumann equation,
$$
H\star {\cal W}_{EE'}- {\cal W}_{EE'} \star H (x,p)= i\partial _t {\cal W}_{EE'}.
$$
To compare with Wofgang's Summary D.5, first take into consideration our somewhat dyslexic adherence to Moyal's notation which reverses Schleich's E' E ordering; and further recall our stationary equations above amount to just
$$
W_{EE'} \star H (x,p) + H\star W_{EE'}\\ = \Biggl ( H \left (\left( x+{i\over 2} \overrightarrow{\partial }_p \right) ,\left( p-{i\over 2} \overrightarrow{\partial }_x \right) \right ) + H \left (\left( x-{i\over 2} \overrightarrow{\partial }_p \right) ,\left( p+{i\over 2} \overrightarrow{\partial }_x \right) \right ) \Biggr ) W_{EE'}(x,p)\\ =(E+E') W_{EE'},
$$
and the corresponding one for the (-) equation that enters the Moyal evolution equation in phase space,
$$
\Biggl (- H \left (\left( x+{i\over 2} \overrightarrow{\partial }_p \right) ,\left( p-{i\over 2} \overrightarrow{\partial }_x \right) \right ) + H \left (\left( x-{i\over 2} \overrightarrow{\partial }_p \right) ,\left( p+{i\over 2} \overrightarrow{\partial }_x \right) \right ) \Biggr ) W_{EE'}(x,p)\\ =(E-E') W_{EE'}.
$$
A ready check is looking at the fate of the kinetic term quadratic in the ps, etc... The fastidious practice of resolving the translation operators through infinite (Taylor) sums started with Wigner and has not abandoned some educators.
- The takeaway, emphasized by Fairlie in 1964, is that Moyal's equation does not suffice to determine the Wigner function completely, but relies on the above (+) equation as well. This has its analog in Hilbert space: solutions ρ of the von Neumann equation are not fully determined, unless their anticommutator with the hamiltonian is also fixed by an eigenvalue (+) equation.
