Inertia-tensor acceleration of a steadily rotating stellar pattern ($\frac{1}{2}\ddot{I}={\Omega }^2\mathrm{diag}(I_{22}-I_{11},I_{11}-I_{22},0)$)

Use the position second-moment tensor $I_{ij}=\int \rho x_i{x}_j{d}^{3}x$, not the engineering moment-of-inertia convention. If a density pattern rotates steadily about its third principal axis, $I(t)=R(\Omega t)I(0)R(\Omega t{)}^T$. Differentiating twice at a principal-axis instant gives the displayed tensor. The tensor virial theorem then determines the pattern speed from the difference of its first two diagonal components. This density-pattern speed is distinct from stellar mean streaming; in the axisymmetric limit $I_{11}=I_{22}$, this difference cannot determine it.