tokens: of the inner binary progressively suppresses the effect. This is because a in decreases, bringing down the ratio in equation ( 38 ). The exact point at which K-L becomes effective also depends on the mass ratio of the intruder with respect to the inner binary, and on the eccentricities of the inner versus the outer binary. Note that as e in increases, the ratio decreases. Therefore, the K-L mechanism might eventually self-regulate itself: as it grows the inner binary more eccentric, it also makes relativistic precession more effective. The condition might therefore not be satisfied any longer, thus suppressing the K-L effect and eventually altering the overall dynamics of the system. Although our investigation is similar in spirit to that of Hoffman Loeb ( 2007 ), the above discussion highlights the importance of the differences in the two implementations. In particular, Hoffman Loeb ( 2007 ) did not include 1PN and 2PN relativistic precession in their equations. This might significantly alter the overall statistical properties of merging binaries (e.g. eccentricities, coalescence time-scales) because, as we just discussed, the conditions for triggering K-L resonances, and thus the general dynamics of the system, are different. Moreover, our treatment of the hardening in the stellar background is more accurate, since it also reproduces the eccentricity evolution in the hardening phase. This is important because (i), it affects the triggering point of K-L cycles (equation 38 ), and (ii), it has a strong impact on the stability of the hierarchical triplet and on the probability of close encounters leading to chaotic behaviour. 5.2 Astrophysical implications The main aim of this paper is to present and validate the code, and a thorough analysis of the dynamics of MBH triplets is deferred to forthcoming papers of the series we planned. Still, the few cases examined here already provide some interesting astrophysical insights that we briefly outline in the following. In all our simulations, two of the MBHs (not necessarily those originally forming the inner binary) coalesce in less than 1Gyr from the start of the dynamical friction phase of the intruder. Therefore, triple interactions might provide a viable channel to merge MBH binaries in massive, low-density galaxies, where hardening against the stellar and gaseous background might act on a time-scale of several Gyr (Khan et al. 2011 ; Sesana Khan 2015 ; Vasiliev et al. 2015 ). At least a fraction of massive elliptical galaxies, which host the most massive binaries targeted by PTAs, might realistically undergo multiple mergers at z ≲1 (see Gerosa Sesana 2015 , and references therein). If the typical coalescence time-scale of the formed MBHBs is several Gyr, then the occurrence of a second merger will bring in an intruder, typically leading to coalescence on a much shorter time-scale. Triple interactions might therefore be an important channel for merging very massive, gas-poor low-redshift
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