We generalize the concept of Lie triple algebra, introduced as tangent algebra of geodesic homogeneous left Lie loop [19], to some algebraic systems equipped with some more multilinear operations, under an idea based on purely geometric point of view. That is, the operations of Lie triple algebras defined by the parallel torsion and curvature tensors of the canonical connection of homogeneous left Lie loops will be extended to ones defined by some connection whose torsion is not assumed to be parallel. The new algebraic system thus obtained will be called Lie triple multi-algebra. We present a method of constructing a Lie triple multi-algebra from double Lie algebras.