  The Lie subgroup ${\rm GL}^{+}(2,\mathbb{C})$ of all matrices in the Lie
group ${\rm GL}(2,\mathbb{C})$ with positive real determinant is equipped with
a left-invariant sub-Lorentzian (anti)metric defined by the natural structure
of the 4-dimensional Minkowski space-time on the subspace of Hermitian matrices
in its Lie algebra. In base of the corresponding time-anti-optimal control
problem, formulated in the paper, and Pontryagin minimum principle for it,
using geodesics and shortest arcs of the corresponding left-invariant
sub-Riemannian metric on the Lie subgroup ${\rm SL}(2,\mathbb{C})$, the authors
found sub-Lorentzian nonspacelike geodesics and longest arcs.
