Separation of the coupling matrix: $C^{(\varsigma)}(r,\theta) = R^{(\varsigma)}(r) + i\,\Theta^{(\varsigma)}(\theta)$

Let $C^{(\varsigma)} = (\det X)\,X^{-1}\bigl(c^{(\varsigma)}I - B^{(\varsigma)}\bigr)(X^*)^{-1}$ be the coupling matrix of Hintz eq. (7), built from the integrating factor $X$ of eq. (5) and the coefficients $c^{(\varsigma)}, B^{(\varsigma)}$ of eq. (4), for a subextremal Kerr–Newman background, real frequency $\omega$, real azimuthal number $m$ and real spin parameter $\varsigma$ (the paper's two spin systems are $\varsigma = \pm 1$). Let $R^{(\varsigma)}(r) = \begin{pmatrix} R_0 & R_1 \\ -R_1 & R_0\end{pmatrix}$ and $\Theta^{(\varsigma)}(\theta) = \begin{pmatrix} \Theta_0 & \Theta_1 \\ \Theta_1 & -\Theta_0\end{pmatrix}$ be the matrices with the explicit coefficients

Then at every point $(r,\theta)\in U$

This is the separation of variables that makes eq. (8) of the paper possible: the $r$-dependence and the $\theta$-dependence of the coupling split additively. The definitions of $R$ and $\Theta$ are independent of $C$; the theorem is the entrywise coefficient verification (four entry lemmas).