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 
equipped with a multilinear ternary operation ![{\displaystyle [\cdot ,\cdot ,\cdot ]:L\times L\times L\to L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d29bbe58d51a5cada75e7523aa8f0435d03e0cbc)
) satisfying the following conditions (the conditions are stated to mimic left-normed Lie products):
![{\displaystyle [u,v,w]=-[v,u,w]\ \forall \ u,v,w\in L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f190ff3b848d43f768ca1746ce98466df036f0e)
![{\displaystyle [u,v,w]+[v,w,u]+[w,u,v]=0\ \forall \ u,v,w\in L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9466ffebb5b49a8a6e9a87006a782b4e504b04f5)
![{\displaystyle [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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dae97e583c5771c5cc3d3667f9f444515000ac85)
Note that for any Lie ring, defining ![{\displaystyle [a,b,c]:=[[a,b],c]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f179a864e0bc0d5decebf8165e0dc0d167a692a3)
yields a Lie triple system.