Context : The appearance of the B [ e ] phenomenon in evolved massive stars such as B [ e ] supergiants is still a mystery . While these stars are generally found to have disks that are cool and dense enough for efficient molecule and dust condensation , the origin of the disk material is still unclear . Aims : We aim at studying the kinematics and origin of the disk in the eccentric binary system GG Car , whose primary component is proposed to be a B [ e ] supergiant . Methods : Based on medium- and high-resolution near-infrared spectra we analyzed the CO-band emission detected from GG Car . The complete CO-band structure delivers information on the density and temperature of the emitting region , and the detectable \@element [ ] [ 13 ] [ ] [ ] { \mathrm { CO } } bands allow us to constrain the evolutionary phase . In addition , the kinematics of the CO gas can be extracted from the shape of the first \@element [ ] [ 12 ] [ ] [ ] { \mathrm { CO } } band head . Results : We find that the CO gas is located in a ring surrounding the eccentric binary system , and its kinematics agrees with Keplerian rotation with a velocity , projected to the line of sight , of 80 \pm 1 km s ^ { -1 } . The CO ring has a column density of ( 5 \pm 3 ) \times 10 ^ { 21 } cm ^ { -2 } and a temperature of 3200 \pm 500 K. In addition , the material is chemically enriched in \@element [ ] [ 13 ] [ ] [ ] { \mathrm { C } } , which agrees with the primary component being slightly evolved off the main sequence . We discuss two possible scenarios for the origin of the circumbinary disk : ( i ) non-conservative Roche lobe overflow , and ( ii ) the possibility that the progenitor of the primary component could have been a classical Be star . Neither can be firmly excluded , but for Roche lobe overflow to occur , a combination of stellar and orbital parameter extrema would be required . Conclusions :