= Solution
Use rotating <Basis vector>[basis vectors] with $\dot{\mathbf e}_1=\Omega\mathbf e_2$, $\dot{\mathbf e}_2=-\Omega\mathbf e_1$. The inertial <velocity> is $(\dot x_1-\Omega x_2)\mathbf e_1+(\dot x_2+\Omega x_1)\mathbf e_2$. The two fixed rotating anchors are $(0,\pm a)$, so the spring lengths are $\ell_\pm=\sqrt{x_1^2+(x_2\mp a)^2}$. Combining kinetic and elastic energies gives
$$
\boxed{\mathcal L=\frac m2[(\dot x_1-\Omega x_2)^2+(\dot x_2+\Omega x_1)^2]-\frac k2[(\ell_+-l_0)^2+(\ell_--l_0)^2].}
$$
A constant elastic offset has no dynamical effect.
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