  Context: Protostellar jets in high-mass young stellar objects (HMYSOs) play a
key role in the understanding of star formation and provide us with an
excellent tool to study fundamental properties of HMYSOs.
  Aims: We aim at studying the physical and kinematic properties of the near-IR
(NIR) jet of IRAS\,13481-6124 from au to parsec scales.
  Methods: Our study includes NIR data from the Very Large Telescope
instruments SINFONI, CRIRES, and ISAAC. Information about the source and its
immediate environment is retrieved with SINFONI. The technique of
spectro-astrometry is performed with CRIRES to study the jet on au scales. The
parsec-scale jet and its kinematic and dynamic properties are investigated
using ISAAC.
  Results: The SINFONI spectra in the $H$ and $K$ band are rich in emission
lines that are mainly associated with ejection and accretion processes.
Spectro-astrometry is applied to the Br$\gamma$ line, and for the first time,
to the Br$\alpha$ line, revealing their jet origin with milliarcsecond-scale
photocentre displacements ($11-15$\,au). This allows us to constrain the
kinematics of the au-scale jet and to derive its position angle
($\sim216^{\circ}$). ISAAC spectroscopy reveals H$_2$ emission along the
parsec-scale jet, which allows us to infer kinematic and dynamic properties of
the NIR parsec-scale jet. The mass-loss rate inferred for the NIR jet is
$\dot{M}_\mathrm{ejec}\sim10^{-4}\mathrm{\,M_\odot\,yr^{-1}}$ and the thrust is
$\dot{P}\sim10^{-2}\mathrm{\,M_\odot\,yr^{-1}\,km\,s^{-1}}$ , which is roughly
constant for the formation history of the young star. A tentative estimate of
the ionisation fraction is derived for the massive jet by comparing the radio
and NIR mass-loss rates. An ionisation fraction $\lesssim8\%$ is obtained,
which means that the bulk of the ejecta is traced by the NIR jet and that the
radio jet only delineates a small portion of it.
