Geodesic loops with non-solvable left translation groups on $3$-dimensional reductive spaces

There are precisely two classes of the connected almost differentiable simple left A-loops $L$ having dimension $3$ such that the group $G$ generated by the left translations of $L$ is a non-solvable Lie group. The class ${\cal C}_1$ consists of loops having the simple Lie group $G=PSL_2(\mathbb C)$ as the group topologically generated by their left translation. Any loop in this class can be represented by a real parameter $a$. For all real $a$ the loops $L_a$ and $L_{-a}$ are isomorphic. These two loops form a full isotopism class. Any loop $L_a$ with $a \ge 0$ is isomorphic to the geodesic loop of the reductive homogeneous space $G/H$ with respect to the reductive complement $T_1[\sigma _a (G/H)]$ and the corresponding canonical invariant connection. The other class ${\cal C}_2$ of loops consists of $3$-dimensional connected differentiable left A-loops such that the group $G=PSL_2(\mathbb R) \ltimes \mathbb R^3$ is the group topologically generated by the left translations The loops in this class can be represented by two real parameters $a,b$ and form precisely two isomorphism classes, which coincide with the isotopism classes. In the one isomorphism class are the Bruck loops $L_{a,0}$, $a \in \mathbb R$ and the pseudo-euclidean space loop $L_{0,0}$ may be chosen as a representative of this isomorphism class. The other isomorphism class containing the loops $L_{a,b}$ with $b \neq 0$ has as a representative the loop $L_{0,1}$. The loops ${\hat L}_0$ and ${\hat L}_1$ are realized on the pseudo-euclidean affine space. The non-simple $3$-dimensional almost differentiable left A-loops are either the direct products of a $1$-dimensional Lie group with a $2$-dimensional left A-loop isomorphic to the hyperbolic plane loop or the unique Scheerer extension of the Lie group $SO_2(\mathbb R)$ by the $2$-dimensional left A-loop isomorphic to the hyperbolic plane loop.