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Lie and Jordan Triple system have arity 3. A Jordan triple system is a vector space with additional structure is given by a triplet $$(x,y,z)\rightarrow \{x,y,z\}$$ that satisfies the identities $$\{u,v,w\} = \{u,w,v\}$$ and $$\{u,v,\{w,x,y\}\} = \{w,x,\{u,v,y\}\} + \{w, \{u,v,x\},y\} -\{\{v,u,w\},x,y\}.$$ See link text. Every Jordan algebra can be embedded in a Jordan triple system but the converse is not true. Any Jordan triple system is a Lie triple system with respect to the product $$[u,v,w] = \{u,v,w,\} − \{v,u,w\}. $$ The structure of a Lie triple system is given by a bracket satisfying the identities $$ [u,v,w] = − [v,u,w], \qquad [u,v,w] + [w,u,v] + [v,w,u] = 0$$ and $$ [u,v,[w,x,y]] = [[u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y]]. $$