= Solution
Take $ck\gg\Omega$, retaining a finite low-frequency branch and a nonzero <wavevector>. In the <dispersion relation>, neglect $\omega^4$ against $c^2k^2\omega^2$ and neglect $\Omega^2$ against $c^2k^2$ in their coefficient. The <inertial wave> limit is
$$
\boxed{\omega^2\simeq\Omega^2\frac{k_z^2}{k^2}.}
$$
Choose the signed branches $\omega=\epsilon\Omega k_z/k$, $\epsilon=\pm1$. The <phase velocity> normal to the phase planes and the <group velocity> are
$$
\mathbf c_p=\frac\omega{k^2}\mathbf k,\qquad \mathbf c_g=\epsilon\Omega\left(\frac{\mathbf e_z}{k}-\frac{k_z\mathbf k}{k^3}\right).
$$
Therefore $\mathbf k\cdot\mathbf c_g=\epsilon\Omega(k_z/k-k_zk^2/k^3)=0$, so
$$
\boxed{\mathbf c_p\cdot\mathbf c_g=0.}
$$
The geometric reason is that this limiting <dispersion relation> is homogeneous of degree zero in $\mathbf k$: its frequency depends on direction, not magnitude. <Euler homogeneous function theorem> gives $\mathbf k\cdot\nabla_{\mathbf k}\omega=0$. Exact orthogonality holds for this <inertial wave> limit; finite compressibility introduces corrections. At a zero-frequency degeneracy the usual propagating-wave interpretation needs a limiting treatment.
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