  There may exist stellar-mass binary black holes (BBH) which merge while
orbiting nearby a supermassive black hole (SMBH). In such a triple system, the
SMBH will modulate the gravitational waveform of the BBH through orbital
Doppler shift and de Sitter precession of the angular momentum. Future
space-based GW observatories focused on the milli- and decihertz band will be
uniquely poised to observe these waveform modulations, as the GW frequency from
stellar-mass BBHs varies slowly in this band while modulation effects
accumulate. In this work, we apply the Fisher information matrix formalism to
estimate how well space-borne GW detectors can measure properties of BBH+SMBH
hierarchical triples using the GW from orbiting BBH. We extend previous work by
considering the more realistic case of an eccentric orbit around the SMBH, and
notably include the effects of orbital pericenter precession. We find that for
detector concepts such as LISA, B-DECIGO, and TianGO, we can extract the SMBH
mass and semimajor axis of the orbit with a fractional uncertainty below the
0.1% level over a wide range of triple system parameters. Furthermore, we find
that the effects of pericenter precession and orbital eccentricity
significantly improve our ability to measure this system. We also find that
while LISA could measure these systems, the decihertz detector concepts
B-DECIGO and TianGO would enable better sensitivity to the triple's parameters.
