  We calculate and present an analytic form of the magneto-optical conductivity
for the gapped low-energy Kane model in two and three dimensions separately.
The two-dimensional case maps onto the $\alpha$-$\mathcal{T}_3$ model at a
particular value of $\alpha=1/\sqrt{3}$. In two dimensions, two chiral sectors
exist, between which there are no optically activated transitions. In three
dimensions, the extra dimension of dispersion mixes the two sectors so that
intra- and inter-sector transitions can occur. The latter type of transition
can be separated out via circular polarizations of light and shows a distinct
signature in the transverse conductivity.
