The timestamps of now-relative bitemporal databases are modeled as growing, shrinking or rectangular regions. The shape of these regions makes it a challenge to design bitemporal operators that (a) are consistent with the point-based interpretation of a temporal database, (b) preserve the identity of the argument timestamps, (c) ensure locality and (d) perform efficiently. We identify the bitemporal split operator as the basic primitive to implement a wide range of advanced now-relative bitemporal operations. The bitemporal split operator splits each tuple of a bitemporal argument relation, such that equality and standard nontemporal algorithms can be used to implement the bitemporal counterparts with the aforementioned properties. Both a native database algorithm and an SQL implementation are provided. Our performance results show that the bitemporal split operator outperforms related approaches by orders of magnitude and scales well.