  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. 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. 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 13CO 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 12CO band head. 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. The CO ring
has a column density of (5\pm 3)x10^21 cm^-2 and a temperature of 3200\pm 500
K. In addition, the material is chemically enriched in 13CO, 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 extremawould be required.
