matrix $F$ introduced in Eq.  as $$\begin{aligned}
F=\begin{pmatrix}
F_{Z\eta} & F_{Z\zeta} & F_{Z\chi} & F_{Z\omega} \\
F_{Z'\eta} & F_{Z'\zeta} & F_{Z'\chi} & F_{Z'\omega} \\
F_{Y\eta} & F_{Y\zeta} & F_{Y\chi} & F_{Y\omega}
\end{pmatrix}\end{aligned}$$ (We assume for the CP-odd sector gauge boson mass matrix, the first, second and third row/column correspond to $Z,Z',Y_I$, respectively.) In the third row, $F_{Y\eta}$ denotes the coefficient of the two-point transition $Y_I^\mu\partial_\mu\eta$ (similar for $F_{Y\zeta},F_{Y\chi},F_{Y\omega}$). Due to CP-conservation there is no two-point transition between $Y_R^\mu$ and the CP-odd scalars, therefore no confusion would arise. The photon field $A^\mu$ does not have two-point transition with scalars. We would like to denote the submatrix formed by the second, third and fourth column of $F$ as $\tilde{F}$ $$\begin{aligned}
\tilde{F}\equiv\begin{pmatrix}
F_{Z\zeta} & F_{Z\chi} & F_{Z\omega} \\
F_{Z'\zeta} & F_{Z'\chi} & F_{Z'\omega} \\
F_{Y\zeta} & F_{Y\chi} & F_{Y\omega}
\end{pmatrix}\end{aligned}$$ Now the application of Eq.  and Eq.  to the CP-odd scalar sector of the SLH leads to $$\begin{aligned}
\begin{pmatrix}
\zeta^m \\ \chi^m \\ \omega^m
\end{pmatrix}
=\mathbb{M}^{-1}_{DV}R\left[
\begin{pmatrix}
F_{Z\eta} \\ F_{Z'\eta} \\ F_{Y\eta}
\end{pmatrix}\eta
+\tilde{F}\begin{pmatrix}
\zeta \\ \chi \\ \omega
\end{pmatrix}
\right]
\label{eq:zetam}\end{aligned}$$ As before the superscript $m$ denotes canonically-normalized mass eigenstate fields. Inverting Eq.  and using Eq.  will lead to $$\begin{aligned}
\begin{pmatrix}
\zeta 