Context : Protostellar jets from intermediate- and high-mass protostars provide an excellent opportunity to understand the mechanisms responsible for intermediate- and high-mass star formation . A crucial question is if they are scaled-up versions of their low-mass counterparts . Such high-mass jets are relatively rare and , usually , they are distant and highly embedded in their parental clouds . The IRAS 20126+4104 molecular jet , driven by a 10 ^ { 4 } L _ { \sun } protostar , represents a suitable target to investigate . Aims : We present here an extensive analysis of this protostellar jet , deriving the kinematical , dynamical , and physical conditions of the H _ { 2 } gas along the flow . Methods : The jet has been investigated by means of near-IR H _ { 2 } and [ Fe ii ] narrow-band imaging , high resolution spectroscopy of the 1-0 S ( 1 ) line ( 2.12 \mu m ) , NIR ( 0.9-2.5 \mu m ) low resolution spectroscopy , along with ISO-SWS and LWS spectra ( from 2.4 to 200 \mu m ) . Results : The flow shows a complex morphology . In addition to the large-scale jet precession presented in previous studies , we detect a small-scale wiggling close to the source , that may indicate the presence of a multiple system . The peak radial velocities of the H _ { 2 } knots range from -42 to -14 km s ^ { -1 } in the blue lobe , and from -8 to 47 km s ^ { -1 } in the red lobe . The low resolution spectra are rich in H _ { 2 } emission , and relatively faint [ Fe ii ] ( NIR ) , [ O i ] and [ C ii ] ( FIR ) emission is observed in the region close to the source . A warm H _ { 2 } gas component has an average excitation temperature that ranges between 2000 K and 2500 K. Additionally , the ISO-SWS spectrum reveals the presence of a cold component ( 520 K ) , that strongly contributes to the radiative cooling of the flow and plays a major role in the dynamics of the flow . The estimated L _ { H _ { 2 } } of the jet is 8.2 \pm 0.7 L _ { \sun } , suggesting that IRAS 20126+4104 has an accretion rate significantly increased compared to low-mass YSOs . This is also supported by the derived mass flux rate from the H _ { 2 } lines ( \dot { M } _ { out } ( H _ { 2 } ) \sim 7.5 \times 10 ^ { -4 } M _ { \sun } yr ^ { -1 } ) . The comparison between the H _ { 2 } and the outflow parameters strongly indicates that the jet is driving , at least partially , the outflow . As already found for low-mass protostellar jets , the measured H _ { 2 } outflow luminosity is tightly related to the source bolometric luminosity . Conclusions : As for few other intermediate- and high-mass protostellar jets in the literature , we conclude that IRAS 20126+4104 jet is a scaled-up version of low-mass protostellar counterparts .