We study the splitting dynamics of giant vortices in dilute Bose-Einstein condensates by numerically integrating the three-dimensional Gross-Pitaevskii equation in time. By taking advantage of tetrahedral tiling in the spatial discretization, we decrease the error and increase the reliability of the numerical method. An extensive survey of vortex splitting patterns is presented for different aspect ratios of the harmonic trapping potential. The discrete rotational symmetries of the splitting patterns that emerge in the time evolution are in good agreement with predictions obtained by solving the prevailing dynamical instabilities from the Bogoliubov equations. Furthermore, we observe intertwining of the split vortices in prolate condensates and a split-and-revival phenomenon in a spherical condensate.