  We present SINFONI/VLT observations of the DG Tauri jet in the [FeII] lines
with 0.15" angular resolution and R=3000 spectral resolution. We observe an
onion-like velocity structure in [FeII] in the blueshifted jet, similar to that
observed in optical lines. High-velocity gas at ~-200 km/s is collimated inside
a half-opening angle of 4 degrees and medium-velocity gas at ~-100 km/s in a
cone with an half-opening angle 14 degrees. Two new axial jet knots are
detected in the blue jet, as well as a more distant bubble with corresponding
counter-bubble. The periodic knot ejection timescale is revised downward to 2.5
yrs. The redshifted jet is detected only beyond 0.7" from the star, yielding
revised constraints on the disk surface density. From comparison to [OI] data
we infer iron depletion of a factor 3 at high velocities and a factor 10 at
speeds below -100 km/s. The mass-fluxes in each of the medium and high-velocity
components of the blueshifted lobe are ~1.6+-0.8x10^-8 Msun/yr, representing
0.02-0.2 of the disk accretion rate. The medium-velocity conical [FeII] flow in
the DG Tau jet is too fast and too narrow to trace photo-evaporated matter from
the disk atmosphere. Both its kinematics and collimation cannot be reproduced
by the X-wind, nor can the "conical magnetospheric wind". The level of Fe gas
phase depletion in the DG Tau medium-velocity component also rules out a
stellar wind and a cocoon ejected sideways from the high-velocity beam. A
quasi-steady centrifugal MHD disk wind ejected over 0.25-1.5 AU and/or episodic
magnetic tower cavities launched from the disk appear as the most plausible
origins for the medium velocity component in the DG Tau jet. The same disk wind
model can also account for the properties of the high-velocity flow, although
alternative origins in magnetospheric and/or stellar winds cannot be excluded
for this component.
