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A Lie algebra $L$ of dimension $n \ge1 $ may be classified, looking forrestrictions of the size on its second integral homology Lie algebra$H_2(L,\mathbb{Z})$, denoted by $M(L)$ and often called Schur multiplier of$L$. In case $L$ is nilpotent, we proved that $\mathrm{dim} \ M(L) \leq\frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$, and worked onthis bound under various perspectives. In the present paper, we estimate theprevious bound for $\mathrm{dim} \ M(L) $ with respect to other inequalities ofthe same nature. Finally, we provide new upper bounds for the Schur multipliersof pairs and triples of nilpotent Lie algebras, by means of certain exactsequences due to Ganea and Stallings in their original form.

Estimations of the low dimensional homology of Lie algebras with large abelian ideals