What is the link between the rotating wave approximation and the algebraic representation of a dynamical system?

In analyzing a system of two coupled oscillators, I noticed a rather interesting correspondence between the so-called “rotating wave approximation” (RWA) for solving differential equations and the structure of the algebraic representation of the matrix form of those equations.

Background

The system of equations under consideration is
$$
frac{d}{dt}
left( begin{array}{c} a a^* b b^* end{array} right)
=
i left(
begin{array}{cccc}
– omega_a & 0 & -g & g
0 & omega_a & -g & g
-g & g & – omega_b & 0
-g & g & 0 & omega_b
end{array}
right)
left( begin{array}{c} a a^* b b^* end{array} right)
tag{$star$} , .
$$
This system is the Hamiltonian version of the system described in a previous question, and can be derived from the Hamiltonian
$$
H = omega_1 a^* a + omega_2 b^* b
– g left( a b – a^* b – a b^* + a^* b^* right)
$$
using the appropriate Hamiltonian equations of motion
$$
dot{a} = -i frac{partial H}{partial a^*}
quad text{and} quad
dot{b} = -i frac{partial H}{partial b^*}
$$
and similarly for $a^*$ and $b^*$.

The parameter $g$ characterizes the strength of coupling between the two oscillators.
Note that if $g=0$, then we have two uncoupled equations
$$
dot a = -i omega_a a quad text{and} quad dot b = -i omega_b b
$$
with solutions
$$
a(t) = a(0) e^{-i omega_a t} quad text{and} quad b(t) = b(0) e^{-i omega_b t}
$$
representing two uncoupled oscillators.

Now note that the matrix in Equation $(star)$ can be written algebraically as
$$
-i sigma_z otimes
left(
g sigma_x + frac{d}{2} sigma_z + frac{s}{2} mathbb{I}
right)
– g (sigma_y otimes sigma_x)
$$
where $s equiv omega_1 + omega_2$ and $d equiv omega_1 – omega_2$.

Rotating wave approximation

The rotating wave approximation (RWA) comes essentially from noting that while the time dependences of e.g. $a$ and $b^*$ have opposite sign and therefore lead to slow oscillation, the time dependences of e.g. $a$ and $b$ have the same sign and lead to fast oscillation.
Therefore, the terms $ab$ and $a^* b^*$ are dropped from the Hamiltonian.
Doing this leads to an equation like $(star)$, but with a matrix given by
$$
-i sigma_z otimes
left(
g sigma_x + frac{d}{2} sigma_z + frac{s}{2} mathbb{I}
right) , ,
$$
i.e. the $-g (sigma_y otimes sigma_x)$ term is gone.

Question

Why is the rotating wave approximation equivalent to dropping the $(sigma_y otimes sigma_x)$ term in the dynamical equation matrix?
What intuitive link is there between the argument we gave for the RWA (or any other argument) and the dropping of the algebraic term $(sigma_y otimes sigma_x)$?

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This question has been psuedo-cross-posted to the Math site in an attempt to get a rigorous and more mathematically oriented answer.