$N$ effectively acquire a small mass given by : $$m_{LL} = m_{LR} M^{-1}_{RR}\, m_{LR}^T
\label{SRNDSeeSaw}$$ Finally, the diagonalization of $m_{LL}$ : $$T^{N*} m_{LL} T^{N\dagger}_{L} = 
{\rm diag}(m_{\nu_1},m_{\nu_2},m_{\nu_3})$$ allows the determination of the masses of the physical neutrinos $m_{\nu_A}$ and enables the physical neutrino states $N' = (\nu_1,\nu_2,\nu_3)$ to be related to the neutrino gauge fields $N = (\nu_e, \nu_\mu, \nu_\tau)$ by $N' = T^N N$.

Taking into account the above conventions, we now proceed to give expressions for the Cabbibo-Kobayashi-Maskawa (CKM) matrix [@CKM] ($V^{CKM}$) and the corresponding lepton analogue, the Maki-Nakawaga-Sakata (MNS) matrix [@MaNaSa] ($V^{MNS}$). Their definitions derive from the charged current interactions : [^3] $$- {g \over \sqrt{2}} W^+_\mu \bar\Psi_U \gamma^\mu P_L \Psi_D \to
- {g \over \sqrt{2}} W^+_\mu \Psi_{U'} \gamma^\mu P_L V^{CKM} \Psi_{D'}$$ $$- {g \over \sqrt{2}} W^-_\mu \bar\Psi_E \gamma^\mu P_L \Psi_{N} \to
- {g \over \sqrt{2}} W^-_\mu \bar\Psi_{E'} \gamma^\mu P_L V^{MNS} \Psi_{N'}$$ that imply : $$V^{CKM} = T^u T^{d\dagger} \qquad
V^{MNS} = T^e T^{N\dagger}
\label{SRNDCKMMNS}$$ In what follows we will assume that the matrices in Eq.  are real. [^4] Thus, we will write $V^{MNS}$ in terms of three rotation matrices : $$V^{MNS} = R_{23} R_{13} R_{12}
\label{MNSRot}$$ given by : $$R_{23} = \left( \matrix{ 1 & 0 