The $\nu^{}_\mu \to \nu^{}_e$ oscillation probability is fully determined by the CP-conserving quantities ${\cal R}^{}_{ij} \equiv {\rm Re} (U^{}_{\mu i} U^{}_{e j} U^*_{\mu j} U^*_{e i})$ and the Jarlskog invariant of CP violation ${\cal J}^{}_\nu \equiv (-1)^{i+j} \hspace{0.05cm} {\rm Im} (U^{}_{\mu i} U^{}_{e j} U^*_{\mu j} U^*_{e i})$ (for $i, j = 1, 2, 3$ and $i < j$), where $U$ is the unitary PMNS lepton flavor mixing matrix. We find that a Pythagoras-like relation ${\cal J}^{2}_\nu = {\cal R}^{}_{12} {\cal R}^{}_{13} + {\cal R}^{}_{12} {\cal R}^{}_{23} + {\cal R}^{}_{13} {\cal R}^{}_{23}$ holds, and it may hopefully offer a novel cross-check of the result of ${\cal J}^{}_\nu$ that will be directly measured in the next-generation long-baseline neutrino oscillation experiments. Terrestrial matter effects are briefly discussed.