The connection algebra of reductive homogeneous spaces

Consider the smooth sections of the tangent bundle of a reductive homogeneous space. This is a vector space over the field of real numbers. The canonical connection acts as a linear binary operator on this vector space, making it an algebra. If we include another binary operator defined as the negative of the torsion, the resulting algebraic structure is a post-Lie-Yamaguti algebra. This structure is closely related to Lie-Yamaguti algebras.