Neutrino Mixing from CP Symmetry

The neutrino mass matrix has remnant CP symmetry expressed in terms of the lepton mixing matrix, and vice versa the remnant CP transformations allow us to reconstruct the mixing matrix. We study the scenario that all the four remnant CP transformations are preserved by the neutrino mass matrix. The most general parameterization of remnant CP transformations is presented. The lepton mixing matrix is completely fixed by the remnant CP, and its explicit form is derived. The necessary and sufficient condition for conserved Dirac CP violating phase is found. If the Klein four flavor symmetry generated by the postulated remnant CP transformations arises from a finite flavor symmetry group, the phenomenologically viable lepton flavor mixing would be the trimaximal pattern, both Dirac CP phase $\delta_{CP}$ and Majorana phase $\alpha_{31}$ are either $0$ or $\pi$ while another Majorana phase $\alpha_{21}$ is a rational multiple of $\pi$. These general results are confirmed to be true in the case that the finite flavor symmetry group is $\Delta(6n^2)$.