  The Kitaev interaction in a honeycomb lattice with higher-spin $S$ has been
one of the central attractions, as it may offer quantum spin liquids. A
microscopic theory showed that when the Hund's coupling at the transition metal
generates $S> \frac{1}{2}$, the spin-orbit coupling at the heavy ligands
provides a route to the Kitaev interaction. However, there have been debates
over its strength compared to other symmetry-allowed interactions.
Investigating the symmetry of the Hamiltonian for general $S$, we show the
magnon energies at two momentum points related by a broken mirror symmetry
reflect the Kitaev interaction when a magnetic field is in the mirror plane.
Applying the symmetry analysis to CrI$_3$ with $S=\frac{3}{2}$ together with
the available angle-dependent ferromagnetic resonance data, we estimate the
Kitaev interaction out of the full Hamiltonian and find that it is
sub-dominant. Our theory can be tested by inelastic neutron scattering on
candidate materials under the proposed magnetic field direction, which will
advance the search for general $S$ Kitaev materials.
