  Some binary black hole systems potentially observable in LISA could be in
orbit around a supermassive black hole (SMBH). The imprint of relativistic
three-body effects on the waveform of the binary can be used to estimate all
the parameters of the triple system, in particular the mass of the SMBH. We
determine the phase shift in the waveform due to the Doppler effect of the SMBH
up to second order in velocity, which breaks a well-known exact degeneracy of
the lowest-order Doppler effect between the mass of the SMBH and its
inclination. We perform several parameter estimations for LISA signals
including this additional dephasing in the wave, showing that one can determine
accurately all parameters of the three-body system. Our results indicate that
one can measure the mass of a $10^8\,$M$_{\odot}$ SMBH with an accuracy better
than $\sim 30\%$ (resp. $\sim 15\%$) by monitoring the waveform of a binary
system whose period around the SMBH is less 100 yr (resp. 20 yr).
