Speciality of Lie-Jordan algebras (Q5936189)

A Lie-Jordan algebra is a vector space \(A\) over a field \(K\), endowed with a binary operation \([\cdot,\cdot]\) and a ternary operation \(\{\cdot,\cdot,\cdot\}\) such that \((A,[\cdot,\cdot])\) is a Lie algebra, \((A,\{\cdot,\cdot,\cdot\})\) is a Jordan triple system, and both operations satisfy certain compatibility conditions. Any associative algebra \(A\) gives rise to a Lie-Jordan algebra, with the same additive structure as \(A\) and binary and ternary operations defined, respectively, by \([x,y]= xy-yx\) and \(\{x,y,z\}= xyz+zyx\). Lie-Jordan algebras which are subalgebras of such a Lie-Jordan algebra derived from an associative algebra are called special. NEWLINENEWLINENEWLINEThe authors prove that every Lie-Jordan algebra over a field of characteristic different from 2 is special. Then they apply this result to prove the associativity of a certain loop constructed by Grishkov.