TITLE: Low dimensional nilpotent Lie algebras
QUESTION [7 upvotes]: This is a reference request question. I would like to know more on the structure of low dimensional nilpotent lie algebras. I heard that up to dimension 6 there are only finitely many isomorphism classes, and every such algebra admits a gradation with only positive degrees (see http://en.wikipedia.org/wiki/Graded_Lie_algebra). 
Do you know of any source where I can find the corresponding proofs?

REPLY [12 votes]: There is indeed lot of works devoted to classification of nilpotent Lie algebras of low dimension (those cited above and many more), with numerous mistakes and omissions. Even worse, all they are using different nomenclature and invariants to classify the algebras, and it is a nontrivial task to compare different lists. Luckily, Willem de Graaf undertook a painstaking task to make an order out of this somewhat messy situation in "Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2", J. Algebra 309 (2007), 640-653 (http://dx.doi.org/10.1016/j.jalgebra.2006.08.006 ); arXiv:math/0511668 . Even better, he provides an algorithm for identifying any given nilpotent Lie algebra with one in his list, and the corresponding code is available as a part of GAP package. He builds on earlier work of Skjelbred-Sund cited above and his own method of identification of Lie algebras by means of Groebner bases.