Sub-Lorentzian extremals defined by an antinorm

We consider a left-invariant (sub-)Lorentzian structure on a Lie group. We assume that this structure is defined by a closed convex salient cone in the corresponding Lie algebra and a continuous antinorm associated with this cone. We derive the Hamiltonian system for (sub-)Lorentzian extremals and give conditions under that normal extremal trajectories keep their causal type. Tangent vectors of abnormal extremal trajectories are either light-like or tangent vectors of sub-Riemannian extremal trajectories for the sub-Riemannian distribution spanned by the cone.