Nonholonomic Lorentzian geometry on some $\mathbb{H}$-type groups

We consider examples of the $\mathbb{H}$-type groups with the natural horizontal distribution generated by the commutation relations of the group. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1 instead of a positive definite quadratic form. The causal character is defined. We study the reachable set by timelike future directed curves. The parametric equations of geodesics are obtained.