Bounds on the angles for the parameterization of three neutrino mixing

An algebraic approach is used to derive symmetry relations for the parameterization of the three neutrino mixing matrix. The symmetry relations imply bounds on the mixing angles. Including a CP violating phase $\delta$, the mixing angles $\theta_{jk}$ lie in the range [0,$\pi$/2], and $\delta$ lies within [0,$2\pi$). If one restricts the CP phase $\delta \in [0,\pi)$, then one must extend the range of either $\theta_{12}$ {\it or} $\theta_{23}$ to $[0, \pi)$. In particular, for no CP violation, one can set $\delta=0$ {\it and} $\delta=\pi$ with $\theta_{jk}$ in the range [0,$\pi$/2]. Alternatively, one can set $\delta=0$ only, and allow either $\theta_{12}$ {\it or} $\theta_{23}$ to lie within [0,$\pi$).