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We study a known filtration of the second cohomology of a finite dimensionalnilpotent Lie algebra $\mathfrak{g}$ with coefficients in a finite dimensionalnilpotent $\mathfrak{g}$-module $M$, that is based upon a refinement of thecorrespondence between $\mathrm{H}^2(\mathfrak{g},M)$ and equivalence classesof abelian extensions of $\mathfrak{g}$ by $M$. We give a differentcharacterization of this filtration and as a corollary, we obtain an expressionfor the second Betti number of $\mathfrak{g}$. Using this expression, we findbounds for the second Betti number and derive a cohomological criterium for theexistence of certain central extensions of $\mathfrak{g}$.

On a filtration of the second cohomology of nilpotent Lie algebras