Geodesics in the Engel group with a sub-Lorentzian metric

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.