A triple system is a vector space over a field together with a -trilinear mapping . A vector space with triple product is said to be a Lie triple system if (a1),(a2),(a3) for all Setting , then (a3) means that the left endomorphism is a derivation of (cf. also Derivation in a ring). Thus one denotes by . Let be a Lie triple system and let be the vector space of the direct sum of and . Then is a Z2-graded Lie algebra with respect to the product.This algebra is called the standard embedding Lie algebra associated with the Lie triple system . This implies that is a homogeneous symmetric space (cf. also Homogeneous space; Symmetric space), that is, it is important in the correspondence with geometric phenomena and algebraic systems. The relationship between Riemannian globally symmetric spaces and Lie triple systems is given in [a4], and the relationship between totally geodesic submanifolds and Lie triple systems is given in [a1]. A general consideration of supertriple systems is given in [a2] and [a5]. Conversely, given a Lie algebra g with such a symmetric decomposition (i.e., it is the Lie algebra of a symmetric space), the triple bracket [[u, v], w] makes m into a Lie triple system.