Geodesics and curvatures of special sub-Riemannian metrics on Lie groups

Let $G$ be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space $M=G/K$ with the stabilizer $K$; $p\colon G\to G/K=M$ the canonical projection which is a Riemannian submersion for some $G$-left invariant and $K$-right invariant Riemannian metric on $G$, and $d$ is a (unique) sub-Riemannian metric on $G$ defined by this metric and the horizontal distribution of the Riemannian submersion $p$. It is proved that each geodesic in $(G,d)$ is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for $M=G/K$, the author found the curvatures of the homogeneous sub-Riemannian manifold $(G,d)$. In the case $G=\operatorname{Sp}(1)\times\operatorname{Sp}(1)$ with the Riemannian symmetric space $S^3=\operatorname{Sp}(1)=G/\operatorname{diag}(\operatorname{Sp}(1)\times\operatorname{Sp}(1))$ the curvatures and torsions are calculated of images in $S^3$ of all geodesics on $(G,d)$ with respect to $p$.