\sim 
M^{-1}_{\nu_3}
\left(\matrix{
\lambda_{13}^2 &
\lambda_{13}\lambda_{23} &
\lambda_{13}\lambda_{33} \cr
\lambda_{13}\lambda_{23} &
\lambda_{23}^2 &
\lambda_{23}\lambda_{33} \cr
\lambda_{13}\lambda_{33} &
\lambda_{23}\lambda_{33} &
\lambda_{33}^2
}\right)+
M^{-1}_{\nu_2}
\left(\matrix{
\lambda_{12}^2 & 
\lambda_{12}\lambda_{22} &
\lambda_{12}\lambda_{32} \cr
\lambda_{12}\lambda_{22} &
\lambda_{22}^2 &
\lambda_{22}\lambda_{32} \cr
\lambda_{12}\lambda_{32} &
\lambda_{22}\lambda_{32} &
\lambda_{32}^2
}\right)
\label{SRNDmLLM2M3}$$

Given that we assumed SRHND by the $\nu^c_\tau$ neutrino, it follows that the contributions to the 23 block of $m_{LL}$ in Eq.  arising from the terms proportional to $M^{-1}_{\nu_3}$ dominate over the ones proportional to $M^{-1}_{\nu_2}$. [^9] Clearly, the rotations $R_{12}$, $R_{13}$ parameterised by the angles in Eq.  diagonalize $m_{LL}$ in Eq.  up to terms of order ${\cal O}(M^{-1}_{\nu_2})$. Thus the new primed matrices $m^{\prime}_{LL}$ and $m^{\prime\prime}_{LL}$ are given by : $$\begin{aligned}
m_{LL}^{\prime} &\sim& 
M^{-1}_{\nu_3}
\left(\matrix{
\lambda_{13}^2 &
0 &
\lambda_{13} A \cr
0 &
0 &
0 \cr
\lambda_{13} A &
0 &
A^2
}\right)+
M^{-1}_{\nu_2}
\left(\matrix{
\lambda_{12}^2 &
\lambda_{12} {C^2 \over A} &
\lambda_{12} {D^2 \over A} \cr
\lambda_{12} {C^2 \over A} &
 {C^4 \over A^2} &
 {C^2 D^2  \over A^2} \cr
\lambda_{12} {D^2 \over A} &
 {C^2 D^2 \over A^2} &
 {D^4 \over A^2}
}\right) \\
& & 
\nonumber \\
m_{LL}^{\prime\prime} &\sim&
M^{-1}_{\nu_3}
\left(\matrix{
0 & 0 & 0 \cr
0 & 0 & 0 