Consider two coupled oscillators with position coordinates $X_a$ and $X_b$. In general, the motion is described by a system of coupled first order linear differential equations: $$ \frac{d}{dt} \begin{pmatrix} a \\ b \\ a^* \\ b^* \end{pmatrix} = -i \begin{bmatrix} \omega_a & g & 0 & -g \\ g & \omega_b & -g & 0 \\ 0 & g & -\omega_a & -g \\ g & 0 & -g & -\omega_b \end{bmatrix} \begin{pmatrix} a \\ b \\ a^* \\ b^* \end{pmatrix} $$ where $a \equiv X_a + i \dot X_a$ and similarly for $b$. It is common in the analysis of this problem to drop the anti-diagonal terms in the matrix, thus decoupling the upper left and lower right blocks. Dropping the anti-diagonal terms is called the "rotating wave approximation" (RWA) and is supposedly good when $g \ll \omega_a, \omega_b$.
Why is the RWA justified?
Physically, dropping the anti-diagonal means that the variables $a$ and $b$ which would be purely clockwise-rotating in the uncoupled $(g=0)$ case couple only to each other and not to the counterclockwise-rotating terms $a^*$ and $b^*$. Perhaps more interestingly I noticed that the matrix can be written in an algebraic form $^{[a]}$ $$ -i \sigma_z \otimes \left( g \sigma_x + \frac{\Delta}{2} \sigma_z + \frac{S}{2} \mathbb{I} \right) - i g (\sigma_y \otimes \sigma_x) $$ where $\Delta \equiv (\omega_a - \omega_b) / 2$ and $S \equiv (\omega_a + \omega_b) / 2$. In this form, the RWA corresponds precisely to dropping the $-ig(\sigma_y \otimes \sigma_x)$ term. I could imagine that this algebraic representation might help explain why the RWA works.
Another final observation is that the eigenvalues of the matrix change only sightly when using the RWA in the limit of $g \ll \omega_a, \omega_b$, which seems to suggest that the RWA is indeed a good approximation. But still, why does it work?
$[a]$ The $\sigma$'s refer to the Pauli matrices.
This question is more or less a rewrite of question 467342 from the Physics site. Here I'm trying to draw more specific attention to the question of why the RWA works at all, and to get insight from the mathematics community.