for seesaw} \end{aligned}$$ where ${a_ + } = {s_{23}}{a_2} + {c_{23}}{a_3}$, ${b_ + } = {s_{23}}{b_2} + {c_{23}}{b_3}$, ${a_ - } = {c_{23}}{a_2} - {s_{23}}{a_3}$ and ${b_ - } = {c_{23}}{b_2} - {s_{23}}{b_3}$. There are two more similar relations to Eq.(\[Eq:M\_nu for 23\]) that determine $\theta_{12,13}$ for given $M_{ij}$ [@Useful2] and these two relations give rise to $$\begin{aligned}
&&
\sin 2{\theta _{12}}\left[ {\frac{{c_{13}^2\left( {a_1^2 + b_1^2} \right) - s_{13}^2\left( {a_ + ^2 + b_ + ^2} \right)}{e^{2i{\delta _{CP}}}}}{{\cos 2{\theta _{13}}}} - \left( {a_ - ^2 + b_ - ^2} \right)} \right] =  - 2\cos 2{\theta _{12}}\frac{{{a_1}{a_ - } + {b_1}{b_ - }}}{{{c_{13}}}},
\label{Eq:12 for seesaw}\\
&&
\sin 2{\theta _{13}}\left[ {\left( {a_ + ^2 + b_ + ^2} \right){e^{i{\delta _{CP}}}} - \left( {a_1^2 + b_1^2} \right){e^{ - i{\delta _{CP}}}}} \right] = 2\cos 2{\theta _{13}}\left( {{a_1}{a_ + } + {b_1}{b_ + }} \right).
\label{Eq:13 for seesaw}\end{aligned}$$ Similarly, neutrino masses accompanied by Majorana phases are calculated to be: $$\begin{aligned}
{m_1}{e^{ - i{\phi _1}}} &=& a_ - ^2 + b_ - ^2 - \frac{{{a_1}{a_ - } + {b_1}{b_ - }}}{{{t_{12}}{c_{13}}}},
\nonumber\\
{m_2}{e^{ - i{\phi _2}}} &=& a_ - ^2 + b_ - ^2 + \frac{{{t_{12}}}}{{{c_{13}}}}\left( {{a_1}{a_ - } + {b_1}{b_ - }} \right),
\label{Eq:masses 