Lie triple system

Definition
A Lie triple system is an analogue of Lie ring where we only have ternary operations. Specifically, it is a ternary non-associative ring (an abelian group $$L$$ equipped with a multilinear ternary operation $$[ \cdot, \cdot,\cdot]:L \times L \times L \to L$$) satisfying the following conditions (the conditions are stated to mimic left-normed Lie products):


 * 1) $$[u,v,w] = -[v,u,w] \ \forall \ u,v,w \in L$$
 * 2) $$[u,v,w] + [v,w,u] + [w,u,v] = 0 \ \forall \ u,v,w \in L$$
 * 3) $$[u,v,[w,x,y]] = u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y \ \forall u,v,w,x,y \in L$$

Note that for any Lie ring, defining $$[a,b,c] := [[a,b],c]$$ yields a Lie triple system.