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][1]. 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
$$    [u,v,[w,x,y]] = [[u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y]]. $$


  [1]: http://en.wikipedia.org/wiki/Jordan_triple_system#Jordan_triple_systems