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The authors found geodesics, shortest arcs, cut loci, and conjugate sets for some leftinvariant sub-Riemannian metric on the Lie group (2) that is right-invariant relative to the Lie subgroup (2) ⊂ (2) (in other words, for invariant sub-Riemannian metric on weakly symmetric space ((2) × (2))/(2)).

Geodesics and shortest arcs of a special sub-Riemannian metric on the Lie group SL(2)