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

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