  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 study some properties of horizontal curves on E.
Second, we prove that time-like normal geodesics are locally maximizers in the
Engel group, and calculate the explicit expression of non-space-like geodesics.
