  A Lie algebra $L$ of dimension $n \ge1 $ may be classified, looking for
restrictions 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 on
this bound under various perspectives. In the present paper, we estimate the
previous bound for $\mathrm{dim} \ M(L) $ with respect to other inequalities of
the same nature. Finally, we provide new upper bounds for the Schur multipliers
of pairs and triples of nilpotent Lie algebras, by means of certain exact
sequences due to Ganea and Stallings in their original form.
