対称空間の代数的モデルとしての準群について

In the previous paper [4], we have studied a characterization of linearly connected manifolds with parallel torsion and curvature by their tangent algebras. Lie algebra Lie group correspondence and Lie triple system symmetric space correspondence are found there in the special cases.
On the other hand, as a generalization of Lie group with ( - ) -connection of Cartan, we have a binary-systematic characterization of linearly connected manifold in our minds. From such a view point, we shall try to present in this note a quasigroup, called a symmetric loop, as an algebraic model of symmetric space. In [5], O. Loos has introduced an axiomatic binary system in symmetric space and defined the symmetric space by means of the multiplication. We were motivated by this work to construct the symmetric loop.
At the last part of the present note, the family of all left translations of the symmetric loop will be observed on the lines of Lie triple family of transformations of T.Nono [6].