= Solution
Ignoring the distant planet at first, the inner two-by-two block has <eigenvalues> zero and $-(B_{12}+B_{21})$. Therefore its fast differential <secular eigenmode> has
$$
\boxed{|\lambda_{\rm fast}|\simeq B_{12}(1+L_1/L_2).}
$$
For the slow mode, strong inner coupling makes $y_1\simeq y_2=y_{\rm in}$. Add the first two evolution equations weighted by <angular momenta> and divide by $L_{12}=L_1+L_2$. The <rigid inner pair in hierarchical secular inclination dynamics> has equations
$$
\dot y_{\rm in}=ic(y_3-y_{\rm in}),\qquad
\dot y_3=ic\frac{L_{12}}{L_3}(y_{\rm in}-y_3),
$$
with
$$
c=\frac{L_1B_{13}+L_2B_{23}}{L_{12}}
\simeq B_{13}\frac{L_1/L_2+\alpha^{-3/2}}{1+L_1/L_2}.
$$
The two-plane matrix has <eigenvalues> zero and $-c(1+L_{12}/L_3)$. The architecture implies $L_3\gg L_{12}$ for comparable masses, so its leading slow frequency is
$$
\boxed{|\lambda_{\rm slow}|\simeq B_{13}\frac{L_1/L_2+\alpha^{-3/2}}{1+L_1/L_2}.}
$$
The neglected factor $1+L_{12}/L_3$ gives a relative correction of order $\sqrt\beta$; imperfect locking of the inner pair supplies higher-order corrections in its weak coupling to the outer planet.
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