Solvable Lie algebras derived from Lie hyperalgebras

Recently in \cite{s-n}, we have investigated Lie algebras and abelian Lie algebras derived from Lie hyperalgebras using the fundamental relations $\mathcal{L}$ and $\mathcal{A}$, respectively. In the present paper, continuing this method we obtain solvable Lie algebras from Lie hyperalgebras by $\mathcal{S}_n$-relations. We show that $\bigcap_{n\geq 1}\mathcal{S}^*_n$ is the smallest equivalence relation on a Lie hyperalgebra such that the quotient structure is a solvable Lie algebra. We also provide some necessary and sufficient conditions for transitivity of the relation $\mathcal{S}_n$ using the notion of $\mathcal{S}_n$-part.