= Rigid inner pair in hierarchical secular inclination dynamics
If two nearby inner planets couple much more strongly to each other than to a distant third planet, their slow <secular eigenmode> has a common <complex inclination>. With $L_{12}=L_1+L_2$, the effective inner-to-outer coupling is
$$
c=\frac{L_1B_{13}+L_2B_{23}}{L_{12}}.
$$
<Angular momentum> balance makes the reverse coupling $cL_{12}/L_3$. The reduced matrix has <eigenvalues> $0$ and $-c(1+L_{12}/L_3)$. If $L_3\gg L_{12}$, the nonzero frequency is approximately $-c$, while the fast inner differential mode has frequency $-(B_{12}+B_{21})$.
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