  For a non-abelian Lie algebra $L$ of dimension $n$ with the derived
subalgebra of dimension $m$ , the first author earlier proved that the
dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$.
In the current work, we give some new inequalities on the exterior square and
the Schur multiplier of Lie algebras and then we obtain the class of all
nilpotent Lie algebras which attains the above bound. Moreover, we also improve
this bound as much as possible.
