Isoclinic relative extensions of pairs of lie algebras

In the present paper we introduce the concept of isoclinism on the relative central extensions of pairs of Lie algebras, which forms an equivalence  relation and we show that each equivalence class contains a stem relative central extension. Also, we determine the structure of all relative central extensions in an isoclinism family of a given relative central extension. Moreover, we show that under some conditions, two concepts of isoclinism and isomorphism between the relative central extensions of pairs of nite dimensional Lie algebras are coincide.Key words: Pair of Lie algebras, relative central extension, isoclinism