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?
Classification of nilpotent Lie algebras in characteristic 0 is an old problem, with a lot of literature. For the dimensions up to 6 there is a finite list. Among the many relevant papers on MathSciNet, I'll list just a few:
MR2372566 (2009a:17027) 17B50 (17B20 17B30) Strade, H. (D-HAMBMI) Lie algebras of small dimension. Lie algebras, vertex operator algebras and their applications, 233–265, Contemp. Math., 442, Amer. Math. Soc., Providence, RI, 2007.
MR0498734 (58 #16802) 17B30 Skjelbred, Tor; Sund, Terje Sur la classification des alg`ebres de Lie nilpotentes. (French. English summary) C. R. Acad. Sci. Paris S´er. A-B 286 (1978), no. 5, A241–A242.
MR855573 (87k:17012) 17B30 Magnin, L. (F-DJON-P) Sur les alg`ebres de Lie nilpotentes de dimension 7. (French. English summary) [Nilpotent Lie algebras of dimension 7] J. Geom. Phys. 3 (1986), no. 1, 119–144.
MR1737529 (2001i:17010) 17B30 (17B05) Tsagas, Gr. (GR-THESS-DMP) Classification of nilpotent Lie algebras of dimension eight. J. Inst. Math. Comput. Sci. Math. Ser. 12 (1999), no. 3, 179–183.
EDIT: This is a somewhat random sample (I'm not a specialist), but these papers recall results for low dimensions and have many references to older literature. The reviews in Math Reviews (MathSciNet) are helpful to look at, if you have access. There is also a fairly modern book, which is very high-priced and probably difficult to access:
MR1383588 (97e:17017) Goze, Michel(F-HALS); Khakimdjanov, Yusupdjan(UZ-AOS) Nilpotent Lie algebras. Mathematics and its Applications, 361. Kluwer Academic Publishers Group, Dordrecht, 1996. xvi+336 pp. ISBN: 0-7923-3932-0 17B30 (17-02 17B40 17B56)
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.
Many articles on classification of low-dimensional Lie algebras do contain mistakes. To the best of my knowledge, the full detailed proof is provided in the dissertation of Ming-Peng Gong:
where he classifies all algebras up to dimension 7 over algebraically closed fields of any characteristics except 2, and also over reals.
The paper Invariants of real low dimension Lie algebras (journal link) lists all real Lie algebras of dimension $\leq 5$ and all nilpotent of dimension $\leq 6$ along with its invariants. It also contains references to the papers where the classifications are obtained: work of Mubarakzyanov and Morozov.
In his 1957 paper Dixmier computes the centres of the universal enveloping of all f-d complex Lie algebras up to dimension 5 and, in particular, lists them all.