1. \forall \ell, \exists (2\ell + 1) \times (2\ell + 1) matrices
   \mathbf{J}_x^\ell, \mathbf{J}_y^\ell, \mathbf{J}_z^\ell satisfying commutation relations
   \left[\mathbf{J}_i, \mathbf{J}_j\right] = -i\epsilon_{ijk}\mathbf{J}_k, with
   \mathbf{J}_z \hat{=} \mathrm{diag}(-\ell, -\ell + 1 \ldots \ell - 1, \ell).

for \ell = 1,

\small\mathbf{J}_x \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{bmatrix}; \mathbf{J}_y \hat{=} {1 \over \sqrt{2}}\begin{bmatrix}0 & i & 0 \\ -i & 0 & i \\ 0 & -i & 0\end{bmatrix}; \mathbf{J}_z \hat{=} \begin{bmatrix}-1 & 0 & 0 \\ 0 &0 &0 \\ 0& 0 &1\end{bmatrix}.

2. For fixed m (to leading order):

\left(-\mathbf{\Omega}_0^2 + 2 \omega m \mathbf{R} + \omega^2 \mathbf{\Delta}\right)\mathbf{c} = 0

For fixed n (to leading order):

(-\omega_0^2 \mathbb{1}_{2\ell+1} + 2 \omega \mathbf{J}_z R_{n,n} + \omega^2 \mathbb{1}_{2\ell+1})\mathbf{y} = 0

3. Under separation of variables, \vec{\xi}_{n\ell m}(r, \theta, \varphi) = \vec{\xi}_{n\ell}(r) Y_\ell^m(\theta, \phi) \iff \underbrace{\tilde{R}_{n, n', m, m'} = R_{n,n'} J_{m,m'}}_{\text{this is a }\textbf{tensor product}!}

\implies \left(-\mathbf{\Omega_0}^2 \otimes \mathbb{1}_{2\ell+1} + 2 \omega \underbrace{\mathbf{R} \otimes \mathbf{J}_z}_{\tilde{\mathbf{R}}} + \omega^2 \mathbf{\Delta} \otimes \mathbb{1}_{2\ell+1}\right)\mathbf{x} = 0

4. Let’s associate with each mass shell at r both \Omega(r)
   (as is customary), and also a rotational axis
   \hat{\mathbf{n}}(r) =\sum_i n_i \mathbf{e}_i.

\small \begin{aligned} \mathbf{R}_{n\ell, n\ell} &= b_{n\ell}\int {\mathbf{d}^\ell}^\dagger(\beta(r)) \Omega(r) \mathbf{J}_z {\mathbf{d}^\ell}(\beta(r))\ K(r)\ \mathrm{d} r\\ &= b_{n\ell}\int \Omega(r) (\hat{\mathbf{n}} \cdot \vec{\mathbf{J}})\ K(r)\ \mathrm{d} r\\ &= \boxed{b_{n\ell}\left(\int \vec{\mathbf{\Omega}}(r) K(r)\ \mathrm{d} r\right) \cdot \vec{\mathbf{J}}}. \end{aligned}

\implies For each mode, AM matrix is
specified by usual vector addition.

5. We only assume that the pure p- and g-mode solutions
   are separately amenable to separation of variables;
   the mixed-mode eigenfunctions need not be.

\small \left(\begin{bmatrix} {\color{grey}\mathbf{L}_{\pi\pi}} & \mathbf{L}_{\pi\gamma} \\ \mathbf{L}_{\pi\gamma}^T & {\color{red}\mathbf{L}_{\gamma\gamma}} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} + 2 \omega \begin{bmatrix} {\color{forestgreen}\tilde{\mathbf{R}}_\pi} & 0 \\ 0 & {\color{forestgreen}\tilde{\mathbf{R}}_\gamma}\end{bmatrix} + \omega^2 \begin{bmatrix} \mathbb{1} & \mathbf{D} \\ \mathbf{D}^T & \mathbb{1} \end{bmatrix} \otimes \mathbb{1}_{2\ell+1} \right)\mathbf{x} = 0.

Mode visibilities are specified by \mathbf{x}^\dagger(\mathbb{1} \otimes \mathbf{P})\mathbf{x}, where \mathbf{P} is the projection matrix onto m=0 in the observer’s coordinate frame.

\vdots