  We revisit the Jaynes-Cummings and anti-Jaynes-Cummings model through the
lens of Lie theory, aiming to highlight the efficacy of an operator-based
approach for diagonalization. We focus on explicitly delineating the steps from
an underlying abstract supersymmetry, provided by the $u(1 \vert 1)$
superalgebra, into concrete proper states and energies in the laboratory frame.
Additionally, we explore the anisotropic Rabi model possessing an underlying
supersymmetry, provided by the $osp(2 \vert 2)$ superalgebra, in a squeezed
reference frame, where it is possible to approximate its spectral
characteristics by an effective Jaynes--Cummings model. Finally, we identify a
regime for a factorizable anisotropic Rabi model, exhibiting an equally spaced,
double degenerate energy spectrum with a unique ground state energy. Our work
aims to merge mathematical physics with practical quantum optics, underscoring
the critical role of Lie theory.
