Complete classification of six dimensional non-semi simple Lie algebra I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$).
In this paper, in page 7, it stated that: "There is no complete classification of the six-dimensional real Lie algebras. However, all nilpotent six-dimensional Lie algebras are known." I also found this paper: J.Math.Phys. 17 (1976) 986, which lists nilpotent six-dimensional Lie algebras in Table III, p.991.
My question, again, is that: whether complete classification of 6 dimensional non-semi simple Lie algebra is known; for both real/complex non-semi simple Lie algebra of dimension 6? What is the most complete result? (In which paper/ref can I find a table of their Lie algebra?) Is that Table III a complete or incomplete for all real/complex non-semi simple Lie algebra of dimension 6?
Deep appreciation to whom concern and reply. (My question is motivated by a problem in topological field theory.)

 A: I am considering the complex case.
The list you wrote is complete for the NILPOTENT Lie algebras and was first obtained by Morozov in 1958 (paper in Russian). To obtain the calssification of ALL Lie algebras of dimension 6 one has to consider two more case:


*

*non solvable

*solvable but not nilpotent
As for the non solvable case there are only three possibilities:
1) ${\mathfrak sl}_2(\mathbb C)\oplus {\mathfrak sl}_2(\mathbb C)$
2) ${\mathfrak sl}_2(\mathbb C)\ltimes \mathbb C^3$ where the semidirect product is given by the adjoint action;
3) ${\mathfrak sl}_2(\mathbb C)\ltimes {\mathfrak n}_3(\mathbb C)$ where the semidirect product is by a sutable action on the Heisenberg -Lie algebra  ${\mathfrak n}_3(\mathbb C)$ (3-dimensional, nilpotent).
Then you need to classify the solvable ones. This was achieved by Yamaguchi Mem. Fac. Sci. Kyushu Univ. Ser. A 35, pages 341-351 (1981). Some more informations are contained in a paper by Goze-Ancochea Bermudez (1985) where the variety of Lie algebras is studied as an algebraic variety for dimension up to 7. The relevant information connected to Clark's comment, is the number of irreducible components. 
A: There are exactly four non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely
\begin{align*}
\mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\
\mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi}\mathfrak{r}_{3,1}(\Bbb C) \cong \mathfrak{aff}(\Bbb C^2)\cong \mathfrak{gl}_2(\Bbb C)\ltimes \Bbb C^2\\
\mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \mathfrak{n}_3(\Bbb C)\\
(\mathfrak{sl}_2(\Bbb C) &\ltimes_{\phi} \Bbb C^2))\oplus \Bbb C \cong \mathfrak{sl}_2(\Bbb C)\ltimes_{\psi}\Bbb C^3
\end{align*}
The proof is straightforward, by computing all derivation algebras of $3$-dimensional Lie algebras $\mathfrak{r}$, and then considering Lie algebra homomorphisms
$$
\phi\colon \mathfrak{sl}_2(\Bbb C)\rightarrow {\rm Der}(\mathfrak{r}).
$$
The only derivation Lie algebras with are not solvable are the ones of $\Bbb C^3$, of the Heisenberg Lie algebra $\mathfrak{n}_3(\Bbb C)$, and of the solvable non-nilpotent Lie algebra $\mathfrak{r}_{3,1}(\Bbb C)$. Their derivation algebras are isomorphic to $\mathfrak{sl}_3(\Bbb C)\oplus \Bbb C$,  $\mathfrak{aff}(\Bbb C^2)$ and $\mathfrak{aff}(\Bbb C^2)$.
I should add one reference, for the case of real numbers, namely the classification
of all solvable real six-dimensional Lie algebras by Turkowsky:
P. Turkowski. Solvable Lie algebras of dimension six. J. Math. Phys. 31 (1990), 1344-1350.
As mentioned before, the classification of nilpotent Lie algebras of dimension $6$ has been done earlier by Morozow in $1958$, valid for all fields of characteristic zero. Recently this classification has been extended to arbitrary fields, by Serena Cicalo, Willem A de Graaf and Csaba Schneider here:http://arxiv.org/abs/1011.0361.
A: A classification of d-dimensional Lie algebras for d < 7 appears in Appendix of the book
Classification and identification of Lie algebras
L.Snobl and P.~Winternitz, vol. 33, CRM Monograph Series, 2014.
