  A significant number of stellar-mass black-hole (BH) binaries may merge in
galactic nuclei or in the surrounding gas disks. With purposed space-borne
gravitational-wave observatories, we may use such a binary as a signal carrier
to probe modulations induced by a central supermassive BH (SMBH), which further
allows us to place constraints on the SMBH's properties. We show in particular
the de Sitter precession of the inner stellar-mass binary's orbital angular
momentum (AM) around the AM of the outer orbit will be detectable if the
precession period is comparable to the duration of observation, typically a few
years. Once detected, the precession can be combined with the Doppler shift
arising from the outer orbital motion to determine the mass of the SMBH and the
outer orbital separation individually and each with percent-level accuracy. If
we further assume a joint detection by space-borne and ground-based detectors,
the detectability threshold could be extended to a precession period of ~100
yr.
