= Solution
For a uniform axial <magnetic field>, the anisotropic <Shear Alfvén wave> is predominantly horizontally polarized, whereas the <pseudo-Alfvén wave> is predominantly vertically polarized. Without rotation both have characteristic <frequency> $\omega_A\sim|k_\parallel|v_A$. Pressure projection makes the effective <Coriolis acceleration> couple these two transverse <wave polarization> states at the inertial-wave rate
$$
\omega_I\sim2\Omega\frac{|k_\parallel|}{k_\perp}.
$$
Thus the useful comparison is not $\Omega$ alone, nor anisotropy alone, but
$$
\boxed{\frac{\omega_I}{\omega_A}\sim\frac{2\Omega}{k_\perp v_A}\ll1.}
$$
In this limit <rotational coupling of shear and pseudo-Alfvén waves> is negligible on an Alfvén propagation time, so the two familiar linear polarizations are decoupled to leading order. The factor $k_\parallel$ cancels: simply making both disturbances more anisotropic does not by itself remove their relative rotational coupling.
For a nonlinear reduction one must also make the rotational mixing slow compared with the nonlinear evolution being retained. Under Alfvénic <critical balance>, $|k_\parallel|v_A\sim k_\perp U$, the same inequality gives $\omega_I\ll k_\perp U$. Rapid rotation relative to the horizontal nonlinear rate can therefore coexist with leading Alfvénic decoupling in the window
$$
\boxed{k_\perp U\ll2\Omega\ll k_\perp v_A.}
$$
High <electrical conductivity> makes magnetic diffusion slow, and the corresponding inviscid approximation is also assumed. If $2\Omega\gtrsim k_\perp v_A$, rotational and magnetic restoring effects are comparable and the modes are mixed magneto-Coriolis disturbances; in the rotation-dominated limit they separate into fast inertial-like and slow magnetostrophic branches, rather than into independently evolving shear and pseudo-Alfvén polarizations.
At finite amplitude, independence is generally one-way in the usual anisotropic <ideal magnetohydrodynamics> reduction: the Alfvénic subsystem has no leading feedback from pseudo-Alfvénic perturbations, but it can still advect them. The small rotational-coupling condition removes an additional two-way coupling, rather than all nonlinear <advection>.
This is an asymptotic finite-timescale statement. The two nonrotating Alfvén polarizations are degenerate, so even a weak nonzero rotational coupling can rotate their superposition over the much longer time $\omega_I^{-1}$. Exact decoupling for arbitrarily long times does not follow from a small ratio alone. The strictly $k_\parallel=0$ nonpropagating limit is separate.
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