  Astrometric precision and knowledge of the point spread function are key
ingredients for a wide range of astrophysical studies including time-delay
cosmography in which strongly lensed quasar systems are used to determine the
Hubble constant and other cosmological parameters. Astrometric uncertainty on
the positions of the multiply-imaged point sources contributes to the overall
uncertainty in inferred distances and therefore the Hubble constant. Similarly,
knowledge of the wings of the points spread function (PSF) is necessary to
disentangle light from the background sources and the foreground deflector. We
analyze adaptive optics (AO) images of the strong lens system J0659+1629
obtained with the W. M. Keck Observatory using the laser guide star AO system.
We show that by using a reconstructed point spread function we can i) obtain
astrometric precision of $< 1$ milliarcsecond (mas), which is more than
sufficient for time-delay cosmography; and ii) subtract all point-like images
resulting in residuals consistent with the noise level. The method we have
developed is not limited to strong lensing, and is generally applicable to a
wide range of scientific cases that have multiple point sources nearby.
