The leading order effect of the spin of a distant supermassive black hole (SMBH) on the orbit of compact binary revolving around it appears at the 1.5 post Newtonian (PN) expansion through coupling with the inner and outer orbital angular momenta, and makes an oscillating characteristic contribution to the secular evolution of the compact binary eccentricity over a long time scale compared to the Kozai-Lidov dynamics caused by a non-spinning SMBH.The first direct detection of gravitational wave (GW) by the Laser Interferometric Gravitational-wave Observatory (LIGO) [1] is a historical landmark. It provides a new way to explore the Universe, and opens the era of gravitational wave astronomy.Up to now, several GW events produced by the inspiral and subsequent merger of two black holes (BHs) or two neutron stars were reported in [2][3][4][5][6][7][8]. The signal of GW150914 was strong enough to be apparent, without using any waveform model, in the filtered detector strain data. But the other GW signals were so weak that the matched filtering with waveform templates was essential for the detection. The templates adopted by LIGO in searching for GW signals assumed the circular obits for the compact binaries [3][4][5][6][7][8]. This assumption is reasonable for the isolated binary systems because the angular momentum that gravitational waves carry away causes the orbits to circularize faster than they shrink [9].The real environment where binary BHs live in are often complicated and they are mostly living in multiple systems. Hierarchical triple system is the simplest and stable one, where binary BHs might obtain large eccentricity perturbed by the third body through Kozai-Lidov mechanism [10,11]. The formation channel of binary BHs roughly fall into two categories: they are from the remanent of evolvement of isolated stellar binaries in "field" chanel [12], or they are formed by "dynamically" channels through three body encounters in dense star clusters [13]. Triple system of stars are believed to be common in universe [14][15][16][17][18], it is possible that triple BHs form in the end of the evolution of triple star systems. On the other hand, the dynamical formation requires a very high stellar density which are believed to exist within the cores of globular clusters (GCs), and the BHs being more massive than the average stars sink to the center of GC until the the majority of the BHs reside in the cluster core [19]. This "mass segregation" process guaranteed that three body encounters in the core can frequently occur [20], producing binary BHs at high rates [21].Kozai-Lidov mechanism has an important secular effect in hierarchical triple systems, and plays an important role in dynamical evolution of triples. GWs emit-ted by highly eccentric orbits of compact binaries excited from the Kozai-Lidov mechanism might be detectable by , pulsar timing arrays [25][26][27], and also future space-based GW observatories such as LISA [28]. Even though Kozai-Lidov mechanism has been extended to more general cases [29][30][31][3...