I am struggling with a demonstration about Weyl transform properties. In particular, I want to prove $$\tilde{(\hat{A}\hat{B})}(x, p) = \tilde{A}(x,p)\exp(i\Lambda/2\hbar)\tilde{B}(x,p)\,,$$ where $\Lambda$ is the Poisson bracket operator. I will show some passages: \begin{equation} \begin{split} \tilde{(\hat{A}\hat{B})}(x, p) & = \int e^{-ipy/\hbar} \langle x+y/2| \hat{A}\hat{B}|x-y/2\rangle dy \\ & = \int e^{-ipy/\hbar} \langle x+y/2| \hat{A}|z\rangle \langle z|\hat{B}|x-y/2\rangle dy dz \,. \end{split} \end{equation}
I have also derived the expression $$\langle x|\hat{A}|x^{\prime}\rangle = \int dp e^{ip(x-x^\prime)/\hbar} \tilde{A}((1/2)(x-x^\prime), p)\,.$$ Also, substituing it in the previous equation I cannot see where the Poisson bracket operator shows up.