(Sub) Lorentzian extremals

Let's consider the (sub)Lorentzian structure on a Lie group, invariant under left translations. This structure is defined by a subspace in the Lie algebra equipped with a sign-changing quadratic form, which determines the length of tangent vectors to admissible curves. The problem of maximizing the length of admissible curves connecting given points is posed. This formulation can be generalized: the length of the tangent vector is defined using an antinorm. Analogously to Minkowski space, one can speak of the causal type of a tangent vector (timelike, lightlike, or spacelike). We will discuss conditions under which normal extremal trajectories preserve their causal type, while abnormal extremal trajectories are either subriemannian or lightlike.