model is a $10\times10$ matrix, written as $$\begin{aligned}
M_{\nu} = \left(\begin{array}{c|c}M'_{\nu_L}(4\times4)=0 & M_{D}(4\times6) \\\hline M^T_{D}(6\times4) & M_{N_{R}}(6\times6)\end{array}\right) \;\; .\end{aligned}$$

From Eq. (\[A4potential\]) we obtain the $6\times6$ right-handed neutrino mass matrix\
in the $(N^{(1)}_{R}, N^{(2)}_{R}, N^{(3)}_{R}, N_{R3_{1}}, N_{R3_{2}}, N_{R3_{3}})$ basis $$\begin{aligned}
\label{MR}
M_{N_{R}} = \left(\begin{array}{cccccc}M_{1} & 0 & 0 & 0 & 0 & 0 \\0 & 0 & M_{2} & 0 & 0 & 0 \\0 & M_{2} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 2M_{2} & 0 & 0 \\0 & 0 & 0 & 0 & 2M_{2} & 0 \\0 & 0 & 0 & 0 & 0 & 2M_{2} \end{array}\right)  \;\;,\end{aligned}$$ where we define $M_{2} = \frac{1}{2}M_{5} + Y_{S1}V_{S}$. Since the $A_{5}$ breaking scale $V_{S}$ is assumed to be higher than the subsequent breaking scales of the $A_{4}$ and SM gauge symmetries, we ignore the effects of $\langle S_{3} \rangle$ in the following discussion of the derivation of tribimaximal mixings and we will treat these effects as perturbations when we calculate the neutrino mass spectrum.

The Dirac mass terms of the neutrino sector are given by $$\begin{aligned}
L_{Dirac} &=& Y_{1}\left[L_{L1}N^{(1)}_{R}H_{1} + L_{L3}N_{R}^{(1)}H_{3} \right] + Y_{2}L_{L3}N_{R3}H_{1} + L_{L1}N_{R3}\left[Y_{2}H_{3} + Y_{3}\Phi_{3}\right] \nonumber 