\\
\frac{\sqrt{2}}{t_{2\beta}}\xi-\frac{7c_{2\beta}+c_{6\beta}}{6\sqrt{2}s_{2\beta}^3}\xi^3 & -\frac{1}{\sqrt{2}}\xi+\frac{5+3c_{4\beta}}{12\sqrt{2}s_{2\beta}^2}\xi^3 & 1-\frac{5+3c_{4\beta}}{12s_{2\beta}^2}\xi^2 & \frac{2}{3t_{2\beta}}\xi^2 \\
-\sqrt{2}\xi+\frac{5+3c_{4\beta}}{3\sqrt{2}s_{2\beta}^2}\xi^3 & -\frac{2\sqrt{2}}{3t_{2\beta}}\xi^3 & \frac{2}{3t_{2\beta}}\xi^2 & 1
\end{pmatrix} \nonumber \\
& +\ord(\xi^4)
\label{eq:vij}\end{aligned}$$ $$\begin{aligned}
& F= \nonumber \\
& gf\begin{pmatrix}
\frac{1}{\sqrt{2}c_W t_{2\beta}}\xi^2 & -\frac{1}{2\sqrt{2}c_W}\xi^2 & \frac{1}{2c_W}\xi-\frac{5+3c_{4\beta}}{24c_W s_{2\beta}^2}\xi^3 & \frac{1}{3c_W t_{2\beta}}\xi^3 \\
\frac{\rho}{t_{2\beta}}\xi^2 & \frac{\sqrt{2}}{\sqrt{3-t_W^2}}-\frac{1+2c_{2W}}{2\sqrt{2}c_W^2\sqrt{3-t_W^2}}\xi^2 & \kappa\xi-\frac{\kappa (5+3c_{4\beta})}{12s_{2\beta}^2}\xi^3 & -\frac{1}{3c_W^2\sqrt{3-t_W^2}t_{2\beta}}\xi^3 \\
-\xi+\frac{5+3c_{4\beta}}{6s_{2\beta}^2}\xi^3 & -\frac{2}{3t_{2\beta}}\xi^3 & \frac{\sqrt{2}}{3t_{2\beta}}\xi^2 & \frac{1}{\sqrt{2}}
\end{pmatrix} \nonumber \\
& +\ord(\xi^4)
\label{eq:fij}\end{aligned}$$ where we defined $$\begin{aligned}
\rho\equiv\sqrt{\frac{1+2c_{2W}}{1+c_{2W}}},\quad \kappa\equiv\frac{c_{2W}}{2c_W^2\sqrt{3-t_W^2}}\end{aligned}$$ It is obvious from Eq.  that the scalar kinetic terms in the original $\eta,\zeta,\chi,\omega$ are not canonically normalized, and also obvious from Eq.  that there are general vector-scalar two-point transitions. Especially, the two-point $Z\eta$ transition appears at $\ord(\xi^2)$, only one order of $\xi$ relatively suppressed when compared to $Z\chi$ transition. The appearance of these non-canonically normalized kinetic terms and ‘unexpected’[^6] vector-scalar transitions is the exact reason for introducing the systematic procedure in Section \[sec:sys\].

The $\Upsilon$ vector is computed to be $$\Upsilon=\begin{pmatrix}
1+\frac{1}{s_{2\beta}^2}\xi^2+\ord(\xi^4) \\
-\frac{1}{t_{2\beta}}\xi^2+\ord(\xi^4) \\
-\frac{\sqrt{2}}{t_{2\beta}}\xi
-\frac{3-c_{4\beta}}{\sqrt{2}s_{2\beta}^2 t_{2\beta}}\xi^3+\ord(\xi^5) \\
\sqrt{2}\xi+\frac{3-c_{4\beta}}{3\sqrt{2}s_{2\beta}^2}\xi^3+\ord(\xi^5)
\end{pmatrix}$$ The $\mathbb{C}^{dH}$ matrix is computed to be $$\mathbb{C}^{dH}=\begin{pmatrix}
0 & 0 & -\frac{g}{2c_W}+\frac{g(5+3c_{4\beta})}{24c_W s_{2\beta}^2}\xi^2+\ord(\xi^4) & 0 \\
0 & 0 & -\frac{g(1-t_W^2)}{2\sqrt{3-t_W^2}}+\frac{g\kappa (5+3c_{4\beta})}{12s_{2\beta}^2}\xi^2+\ord(\xi^4) & 0 \\
0 & 0 & -\frac{\sqrt{2}g}{3t_{2\beta}}\xi+\frac{g(7c_{2\beta}+c_{6\beta})}{30\sqrt{2}s_{2\beta}^3}\xi^3+\ord(\xi^5) & 0
\end{pmatrix}$$ The $\mathbb{C}^{Hd}$ matrix is computed to 