  Binary-black-hole (BBH) mergers can take place close to a supermassive black
hole (SMBH) while being in a bound orbit around the SMBH. In this paper, we
study such bound triple systems and show that including the strong gravity
effects of describing the SMBH with a Kerr metric can significantly modify the
dynamics, as compared to a Newtonian point particle description of the SMBH. We
extract the dynamics of the system, using a quadrupole approximation to the
tidal forces due to the SMBH. We exhibit how the gyroscope precession is built
into this dynamics, and find the secular Hamiltonian by both averaging over the
inner and outer orbits, the latter being the orbit of the BBH around the SMBH.
We study the long-time-scale dynamics, including the periastron precession and
GW radiation-reaction of the binary system, finding that the strong gravity
effects of the SMBH can enhance the von Zeipel-Lidov-Kozai mechanism, resulting
in more cycles, higher maximum eccentricity, and thereby a shorter merger time,
particularly when the binary is close to, or at, the innermost stable orbit of
the SMBH. We end with an analysis of the peak frequency of the GW emission from
the binary system, highlighting possible observable signatures in the LISA and
ET frequency bands.
