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There are some results on nilpotent Lie algebras $ L $ investigate thestructure of $ L $ rely on the study of its $2$-nilpotent multiplier. It isshowed that the dimension of the $2$-nilpotent multiplier of $ L $ is equal to$ \frac{1}{3} n(n-2)(n-1)+3-s_2(L).$ Characterizing the structure of allnilpotent Lie algebras has been obtained for the case $ s_2(L)=0.$ This paperis devoted to the characterization of all nilpotent Lie algebras when $ 0\leqs_2(L)\leq 6.$ Moreover, we show that which of them are $2$-capable.

Characterizing nilpotent Lie algebras rely on the dimension of their $2$-nilpotent multipliers