Triple Lie systems associated with \((-1, 1)\) algebras (Q2191882)

If \(J\) is a Jordan algebra one can construct a Lie triple system \(T(J)\) be defining on \(J\) the triple product \([x,y,z]:=(xy)z-x(yz)\). For any right alternative algebra \(A\) one can construct \(T_J(A)=T(A^+)\) (where \(A^+\) is the symmetrization algebra). Recall that a \((- 1, 1)\) ring is a nonassociative ring satisfying the conditions \[ 0 = (x, y, z) + (y, z, x) + (z, x, y),\quad 0 = (x, y, z) + (x, z, y) \] for arbitrary elements in the ring. In a work published by \textit{S. V. Pchelintsev} and \textit{I. P. Shestakov} [J. Algebra 423, 54--86 (2015; Zbl 1369.17029)], the relation between \((-1,1)\)-algebras and Jordan algebras are studied. In the same year, a work by V. Zhelyabin and A. S. Zakharov explores the relations between \((-1,1)\)-algebras and Novikov algebras. The present work deals with right alternative \((-1,1)\)-algebras. It studies relations between \((-1,1)\)-algebras and certain Lie triple systems that are called \(TL_K(A)\), related to central isotopes of \((-1,1)\)-algebras. It is proved that a \((-1,1)\)-algebras \(A\) has nilpotent associator ideal if and only if the Lie triple system \(TL_K(A)\) is nilpotent. The question on the solvability and nilpotency of the Lie triple systems \(T_J(A)\) and \(TL_K(A)\) constructed from \(A\) is also studied.