= Solution
Let $\boldsymbol\omega_a=2\boldsymbol\Omega+\nabla\times\mathbf u$ be <absolute vorticity>, where $\boldsymbol\Omega=(0,0,f/2)$ is constant. Let $\rho$ be <density> and $\alpha$ a materially conserved stratifying scalar, so $D\alpha/Dt=0$. For the incompressible density-stratified ideal fluid one may choose the advected <density> itself; in a thermodynamic description one may use an entropy-like scalar with $\rho=\rho(p,\alpha)$. The <Rossby–Ertel potential vorticity> is
$$
\mathcal P_E=\rho^{-1}\boldsymbol\omega_a\cdot\nabla\alpha.
$$
<Ertel's theorem> states that it is conserved following a parcel under inviscid, adiabatic motion with conservative body force and the stated stratifying-scalar condition.
Start from the rotating <momentum> equation
$$
\frac{D\mathbf u}{Dt}+2\boldsymbol\Omega\times\mathbf u=-\rho^{-1}\nabla p-\nabla\Phi.
$$
Here $\Phi$ includes gravity and any conservative centrifugal potential. Taking its curl gives
$$
\frac{D\boldsymbol\omega_a}{Dt}
=(\boldsymbol\omega_a\cdot\nabla)\mathbf u-\boldsymbol\omega_a\nabla\cdot\mathbf u
+\frac{\nabla\rho\times\nabla p}{\rho^2}.
$$
Mass conservation is $D\rho/Dt=-\rho\nabla\cdot\mathbf u$, specializing to $D\rho/Dt=0$ here. Combining these equations yields
$$
\frac D{Dt}\left(\frac{\boldsymbol\omega_a}{\rho}\right)
=\left(\frac{\boldsymbol\omega_a}{\rho}\cdot\nabla\right)\mathbf u
+\frac{\nabla\rho\times\nabla p}{\rho^3}.
$$
Differentiating the material conservation of $\alpha$ gives $D(\partial_i\alpha)/Dt=-(\partial_i u_j)\partial_j\alpha$. Dotting these two equations, the velocity-gradient terms cancel after relabelling their indices. Therefore
$$
\frac{D\mathcal P_E}{Dt}=\frac{(\nabla\rho\times\nabla p)\cdot\nabla\alpha}{\rho^3}.
$$
If $\alpha=\rho$, the triple product vanishes immediately. More generally, an <equation of state> $\rho(p,\alpha)$ makes $\nabla\rho$ a linear combination of $\nabla p$ and $\nabla\alpha$, so it also vanishes. Hence
$$
\boxed{\frac{D\mathcal P_E}{Dt}=0.}
$$
This proves the conservation law and specifies why $\alpha$ must be a suitable stratifying scalar: an arbitrary unrelated passive tracer need not annihilate the baroclinic triple product. No viscosity or scalar diffusion is permitted in this theorem.
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