  Stellar-mass binary black holes (BBHs) may merge in the vicinity of a
supermassive black hole (SMBH). It is suggested that the gravitational-wave
(GW) emitted by a BBH has a high probability to be lensed by the SMBH if the
BBH's orbit around the SMBH (i.e., the outer orbit) has a period of less than a
year and is less than the duration of observation of the BBH by a space-borne
GW observatory. For such a BBH + SMBH triple system, the de Sitter precession
of the BBH's orbital plane is also significant. In this work, we thus study GW
waveforms emitted by the BBH and then modulated by the SMBH due to effects
including Doppler shift, de Sitter precession, and gravitational lensing. We
show specifically that for an outer orbital period of 0.1 yr and an SMBH mass
of $10^7 M_\odot$, there is a 3\%-10\% chance for the standard, strong lensing
signatures to be detectable by space-borne GW detectors such as LISA and/or
TianGO. For more massive lenses ($\gtrsim 10^8 M_\odot$) and more compact outer
orbits with periods <0.1 yr, retro-lensing of the SMBH might also have a
1%-level chance of detection. Furthermore, by combining the lensing effects and
the dynamics of the outer orbit, we find the mass of the central SMBH can be
accurately determined with a fraction error of $\sim 10^{-4}$. This is much
better than the case of static lensing because the degeneracy between the lens'
mass and the source's angular position is lifted by the outer orbital motion.
Including lensing effects also allows the de Sitter precession to be detectable
at a precession period 3 times longer than the case without lensing. Lastly, we
demonstrate that one can check the consistency between the SMBH's mass
determined from the orbital dynamics and the one inferred from gravitational
lensing, which serves as a test on theories behind both phenomena. The
statistical error on the deviation can be constrained to a 1% level.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena