= {t_{13}}{a_ + }$, ${b_ + } =  - {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}$   $a_1$, $b_1$   $a_+$, $b_+$
    4     ${b_1}{e^{-i{\delta _{CP}}}} = {t_{13}}{b_ + }$, ${a_ + } =  - {t_{13}}{a_1}{e^{ - i{\delta _{CP}}}}$                 

  : \[Tab:NH\]Constraints for NH to induce maximal CP violation

   case                                relevant constraint for $\delta _{CP}=\pm \pi/2$                                    real        imaginary
  ------ ------------------------------------------------------------------------------------------------------------ -------------- --------------
    1     ${a_ + } =  - {t_{13}}{a_1}{e^{ - i{\delta _{CP}}}}$, ${b_ + } =  - {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}$   $a_1$, $b_1$   $a_+$, $b_+$
    2                                ${b_ + } =  - {t_{13}}{b_1}{e^{ - i{\delta _{CP}}}}$                                 $b_1$          $b_+$
    3                                ${a_ + } =  - {t_{13}}{a_1}{e^{ - i{\delta _{CP}}}}$                                 $a_1$          $a_+$

  : \[Tab:IH\]Constraints for IH to induce maximal CP violation

If the atmospheric neutrino mixing is maximal as well, $a_+$ and $a_-$ turn out to be ${a_ + } = (\sigma{a_2} + {a_3})/{\sqrt 2}$ and ${a_ - } = \sigma (\sigma{a_2}- {a_3})/{\sqrt 2}$. Therefore, it can be observed that the relation of ${a_3} = -\sigma{a_2}^\ast$ as long as $a_+\neq 0$ and $a_-\neq 0$ ensures the appearance of the imaginary $a_+$ in all focused cases requiring $a_-$ to be real and similarly for $b_{+,-}$. This constraint on $a_{2,3}$ (or $b_{2,3}$) is equivalent to Eqs.(\[Eq:Maximal-CP1\]) and (\[Eq:Maximal-CP2 and ATM\]). In terms 