  We construct the family of bilinear forms gG on R3+1 for which Galilean
boosts and spatial rotations are isometries. The key feature of these bilinear
forms is that they are parametrized by a Galilean invariant vector whose
physical interpretation is rather unclear. At the end of the paper, we
construct the Poisson bracket associated with the (nondegenerate) antisymmetric
part of gG.
