  Given a nilpotent Lie algebra $\mathfrak n$ we construct a spectral sequence
which is derived from a filtration of its Chevalley-Eilenberg differential
complex and converges to the Lie algebra cohomology of $\mathfrak n$. The limit
of this spectral sequence gives a grading for the Lie algebra cohomology,
except for the cohomology groups of degree 0, 1, $\dim \mathfrak n-1$ and
$\dim\mathfrak n$ as we shall prove. We describe the spectral sequence
associated to a nilpotent Lie algebra which is a direct sum of two ideals, one
of them of dimension one, in terms of the spectral sequence of the co-dimension
one ideal. Also, we compute the spectral sequence corresponding to each real
nilpotent Lie algebra of dimension less than or equal to six.
