  Although thousands of galaxy mergers are known, only a handful of
sub-kiloparsec-scale supermassive black hole (SMBH) pairs have been confirmed
so far, leaving a huge gap between the observed and predicted numbers of such
objects. In this work, we present a detailed analysis of the Sloan Digital Sky
Survey optical spectrum and of near-infrared (NIR) diffraction limited imaging
of SDSS~J1431+4358. This object is a local radio-quiet type 2 active galactic
nucleus (AGN) previously selected as a double AGN candidate on the basis of the
double-peaked [OIII] emission line. The NIR adaptive optics-assisted
observations were obtained at the Large Binocular Telescope with the LUCI+FLAO
camera. We found that most of the prominent optical emission lines are
characterized by a double-peaked profile, mainly produced by AGN
photoionization. Our spectroscopical analysis disfavors the hypothesis that the
double-peaked emission lines in the source are the signatures of outflow
kinematics, leaving open the possibility that we are detecting either the
rotation of a single narrow-line region or the presence of two SMBHs orbiting
around a common central potential. The latter scenario is further supported by
the high-spatial resolution NIR imaging: after subtracting the dominant
contribution of the stellar bulge component in the host galaxy, we detect two
faint nuclear sources at r<0.5 kpc projected separation. Interestingly, the two
sources have a position angle consistent with that defined by the two regions
where the [OIII] double peaks most likely originate. Aside from the discovery
of a promising sub-kiloparsec scale dual AGN, our analysis shows the importance
of an appropriate host galaxy subtraction in order to achieve a reliable
estimate of the incidence of dual AGNs at small projected separations.
