$3$-dimensional loops on non-solvable reductive spaces

Abstract

We treat the almost differentiable left A-loops as images of global differentiable sharply transitive sections $\sigma :G/H \to G$ for a Lie group $G$ such that $G/H$ is a reductive homogeneous manifold. In this paper we classify all $3$-dimensional connected strongly left alternative almost differentiable left A-loops $L$, such that for the corresponding section $\sigma :G/H \to G$ the Lie group $G$ is non-solvable.