  Let E be the Engel group and D be a rank 2 bracket generating left invariant
distribution with a Lorentzian metric, which is a nondegenerate metric of index
1. In this paper, we first prove that timelike normal extremals are locally
maximizing. Second, we obtain a parametrization of timelike, spacelike,
lightlike normal extremal trajectories by Jacobi functions. Third, a discrete
symmetry group and its fixed points which are Maxwell points of of timelike and
spacelike normal extremals, are described. An estimate for the cut time (the
time of loss of optimality) on extremal trajectories is derived on this basis.
