Time-reversible ab initio molecular dynamics based on a lossless multichannel decomposition for the integration of the electronic degrees of freedom [Phys. Rev. Lett. 97, 123001 (2006)] is explored. The authors present a lossless time-reversible density matrix molecular dynamics scheme. This approach often allows for stable Hartree-Fock simulations using only one single self-consistent field cycle per time step. They also present a generalization, introducing an additional "forcing" term, that in a special case includes a hybrid Lagrangian, i.e., Car-Parrinello-type, method, which can systematically be constrained to the Born-Oppenheimer potential energy surface by using an increasing number of self-consistency cycles in the nuclear force calculations. Furthermore, in analog to the reversible and symplectic leapfrog or velocity Verlet schemes, where not only the position but also the velocity is propagated, the authors propose a Verlet-type density velocity formalism for time-reversible Born-Oppenheimer molecular dynamics.