  Many triple-star systems have an inner pair with an orbital period of a few
days only. A common mechanism to explain the short-period pile-up present in
the observations is the migration through Lidov-Kozai cycles combined with
tidal friction. Here, we revisit this mechanism and aim to determine the
initial orbital configurations leading to this process. We show that the mutual
inclination of the triple-star system is not the only critical parameter, since
the eccentricity as well as the argument of the pericenter of the inner orbit
also play an important role in the establishment of the Lidov-Kozai migration.
Our framework is the secular hierarchical three-body problem (octupole order
approximation) with general relativity corrections, including the effects of
tides, stellar oblateness and magnetic spin-down braking. Both the orbital and
the spin evolutions are considered. Extensive numerical simulations with
uniform and non-uniform distributions of the initial orbital parameters are
carried out, and unbiased initial conditions leading to Lidov-Kozai migration
are revealed. Finally, we highlight the importance of the initial "Kozai
constant" $h=\sqrt{(1-e^2)}\cos{i}$ in the dynamical evolution of triple-star
systems, by showing that phase portraits at given $h$-values unveil different
evolution paths.
