In this article, we study Lie hyperalgebras and compare them with Lie algebras. We introduce the -relation on a Lie hyperalgebra such that the quotient structure by its transitive closure is a Lie algebra. Also, we provide the required conditions to obtain abelian Lie algebras from Lie hyperalgebras by the -relation. It is shown that the transitive closures of and are the smallest equivalence relations on Lie hyperalgebras whose quotients turn into the fundamental Lie algebras and abelian Lie algebras, respectively. Moreover, some necessary and sufficient conditions for transitivity of the relations are given by using -parts and -parts, equipped with illustrated examples.