containing $M_{ij}$ as elements to yield $M_\nu = -m_D M_R^{-1} m_D^T$, where $m_D$ is a $3 \times 2$ Dirac neutrino mass matrix. We parameterize $M_R$ by $$\begin{aligned}
M_R = \left(
  \begin{array}{cc}
  M_1 & 0   \\ 
  0       & M_2
  \end{array}
\right)\quad (M_1<M_2),\end{aligned}$$ and $m_D$ by $$\begin{aligned}
m_D = 
\left(
  \begin{array}{cc}
    \sqrt{M_1}a_1  & \sqrt{M_2}b_1   \\
    \sqrt{M_1}a_2  & \sqrt{M_2}b_2   \\
    \sqrt{M_1}a_3  & \sqrt{M_2}b_3   \\
  \end{array}
\right),\end{aligned}$$ which result in $$\begin{aligned}
M_\nu &=& \left( \begin{array}{*{20}{c}}
M_{ee}&M_{e\mu }&M_{e\tau }\\
M_{e\mu }&M_{\mu \mu }&M_{\mu \tau }\\
M_{e\tau }&M_{\mu \tau }&M_{\tau \tau }
\end{array} \right)
=
-\left(
  \begin{array}{ccc}
    a_1^2 + b_1^2   & a_1a_2 + b_1b_2   &  a_1a_3 + b_1b_3  \\
    a_1a_2 + b_1b_2 & a_2^2 + b_2^2     &  a_2a_3 + b_2b_3  \\
    a_1a_3 + b_1b_3 & a_2a_3 + b_2b_3   &  a_3^2 + b_3^2   \\
  \end{array}
\right),
\label{Eq:Mnu}\end{aligned}$$ where the minus sign in front of the mass matrix is discarded for the later discussions. One of the masses of $\nu_{1,2,3}$ is required to vanish owing to $\det\left(M_\nu\right)=0$.

The useful relation Eq.(\[Eq:M\_nu for 23\]) expressed in terms of $M_{ij}$ is converted into $$\begin{aligned}
{a_ + }{a_ - } + {b_ + }{b_ - } =  - {t_{13}}\left( {{a_1}{a_ - } + {b_1}{b_ - }} \right){e^{ - i{\delta _{CP}}}},
\label{Eq:23 