  The binary black holes (BBHs) formed near the supermassive black holes
(SMBHs) in the galactic nuclei would undergo eccentricity excitation due to the
gravitational perturbations from the SMBH and therefore merger more
efficiently. In this paper, we study the coupling of the three body 1st
post-Newtonian (PN) effects with the spin effects from the SMBH in the
hierarchical triple system. We extend previous work by including the coupling
between the de Sitter precession and the Lense-Thirring precession from the
SMBH spin. This coupling includes both the precessions of the inner orbit
angular momentum and the Runge-Lenz vector around the outer orbit angular
momentum in a general reference frame. We find the change of the (maximal)
eccentricity in the neighboring Kozai-Lidov cycles due to spin effects is
detectable by LISA in the future. Our general argument on the coupling of the
three body 1PN effects in three body systems could be extended to any other
situation as long as the outer orbital plane evolves.
