Geodesics and shortest arcs of some sub-Riemannian metrics on the Lie groups $SU(1,1)\times \mathbb{R}$ and $S{O}_0(2,1)\times \mathbb{R}$ with three-dimensional generating distributions

We find geodesics, shortest arcs, cut loci, first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups $SU(1,1)\times \mathbb{R}$ and $S{O}_0(2,1)\times \mathbb{R}$.