  The formation of Black Hole (BH) Low-Mass X-ray Binaries (LMXB) poses a
theoretical challenge, as low-mass companions are not expected to survive the
common-envelope scenario with the BH progenitor. Here we propose a formation
mechanism that skips the common-envelope scenario and relies on triple-body
dynamics. We study the evolution of hierarchical triples, following the secular
dynamical evolution up to the octupole-level of approximation, including
general relativity, tidal effects and post-main-sequence evolution, such as
mass loss, changes to stellar radii and supernovae. During the dynamical
evolution of the triple system, the "eccentric Kozai-Lidov" mechanism can cause
large eccentricity excitations in the LMXB progenitor, resulting in three main
BH-LMXB formation channels. Here we define BH-LMXB candidates as systems where
the inner BH companion star crosses its Roche limit. In the "eccentric" channel
(~ 81% of the LMXBs in our simulations), the donor star crosses its Roche limit
during an extreme eccentricity excitation, while still on a wide orbit. Second,
we find a "giant" LMXB channel (~ 11%), where a system undergoes only moderate
eccentricity excitations, but the donor star fills its Roche lobe after
evolving toward the giant branch. Third, we identify a "classical" channel
(~8%), where tidal forces and magnetic braking shrink and circularize the orbit
to short periods, triggering mass transfer. Finally, for the giant channel, we
predict an eccentric ($\sim 0.3-0.6$), preferably inclined (~40, ~140 degreed)
tertiary, typically on a wide enough orbit (~10^4AU), to potentially become
unbound later in the triple evolution.
