$$\begin{aligned}
M_{\nu_L(\nu_{L1}, \nu_{L1})} &=& \frac{Y^2_{1}v^2_{H_1}}{M_1} + \frac{Y^2_{3}(v^2_{\phi_1} + v^2_{\phi_2} + v^2_{\phi_3})}{2M_2} \;\; , \nonumber \\
M_{\nu_L(\nu_{L1}, \nu_{L3_1})} &=& \frac{Y_{2}Y_{3}v_{\phi_1}v_{H_1} + 2Y^2_{3}v_{\phi_2}v_{\phi_3}}{2M_2} \;\; ,  \nonumber \\
M_{\nu_L(\nu_{L1}, \nu_{L3_2})} &=& \frac{Y_{2}Y_{3}v_{\phi_2}v_{H_1} + 2Y^2_{3}v_{\phi_1}v_{\phi_3}}{2M_2} \;\; ,  \nonumber \\
M_{\nu_L(\nu_{L1}, \nu_{L3_3})} &=& \frac{Y_{2}Y_{3}v_{\phi_3}v_{H_1} + 2Y^2_{3}v_{\phi_1}v_{\phi_2}}{2M_2} \;\; ,  \nonumber \\
M_{\nu_L(\nu_{L3_1}, \nu_{L3_1})} &=& \frac{Y^2_{2}v^2_{H_1} + Y^2_{3}(4v^2_{\phi_1} + v^2_{\phi_2} + v^2_{\phi_3})}{2M_2}  \;\; ,  \\ 
M_{\nu_L(\nu_{L3_1}, \nu_{L3_2})} &=& \frac{2Y_{2}Y_{3}v_{H_1}v_{\phi_3} - Y^2_{3}v_{\phi_1}v_{\phi_2}}{2M_2}  \;\; , \nonumber \\
M_{\nu_L(\nu_{L3_1}, \nu_{L3_3})} &=& \frac{2Y_{2}Y_{3}v_{H_1}v_{\phi_2} - Y^2_{3}v_{\phi_1}v_{\phi_3}}{2M_2}  \;\; , \nonumber \\
M_{\nu_L(\nu_{L3_2}, \nu_{L3_2})} &=& \frac{Y^2_{2}v^2_{H_1} + Y^2_{3}(v^2_{\phi_1} + 4v^2_{\phi_2} + v^2_{\phi_3})}{2M_2} \;\; ,  \nonumber \\
M_{\nu_L(\nu_{L3_2}, \nu_{L3_3})} &=& \frac{2Y_{2}Y_{3}v_{H_1}v_{\phi_1} - Y^2_{3}v_{\phi_2}v_{\phi_3}}{2M_2}  \;\; , \nonumber \\
M_{\nu_L(\nu_{L3_3}, \nu_{L3_3})} &=& \frac{Y^2_{2}v^2_{H_1} + Y^2_{3}(v^2_{\phi_1} + v^2_{\phi_2} + 4v^2_{\phi_3})}{2M_2} \;\; . \nonumber  \end{aligned}$$ We can diagonalize the $4\times4$ symmetric mass matrix $M_{\nu_L}$ by writing it in the form $$\begin{aligned}
M_{\nu_L} = \left(\begin{array}{cccc}A & B & C & D \\B & E & F & G \\C & F & H & I \\D & G & I & J\end{array}\right)\end{aligned}$$ in the $(\nu_{\tau'}, \nu_{\tau}, \nu_{\mu}, \nu_{e})$ basis, and then using the unitary transformation, $$\begin{aligned}
M_{diag} = U^{\rm 4g}_{TBM}M_{\nu_L}U^{4g^T}_{TBM} \;\;,\end{aligned}$$ where the 4 generations mixing matrix in the tribimaximal mixing limit can be expressed as $$\begin{aligned}
U^{4g}_{TBM} = \left(\begin{array}{cccc} 