  We have found the algebraic structure of the two-qubit quantum Rabi model
behind the possibility of its novel quasi-exact solutions with finite photon
numbers by analyzing the Hamiltonian in the photon number space. The
quasi-exact eigenstates with at most $1$ photon exist in the whole qubit-photon
coupling regime with constant eigenenergy equal to single photon energy
\hbar\omega, which can be clear demonstrated from the Hamiltonian structure.
With similar method, we find these special "dark states"-like eigenstates
commonly exist for the two-qubit Jaynes-Cummings model, with $E=N\hbar\omega$
(N=-1,0,1,...), and one of them is also the eigenstate of the two-qubit quantum
Rabi model, which may provide some interesting application in a simper way.
Besides, using Bogoliubov operators, we analytically retrieve the solution of
the general two-qubit quantum Rabi model. In this more concise and physical
way, without using Bargmann space, we clearly see how the eigenvalues of the
infinite-dimensional two-qubit quantum Rabi Hamiltonian are determined by
convergent power series, so that the solution can reach arbitrary accuracy
reasonably because of the convergence property.
