Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$ Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From reading this MO post: complete-classification-of-six-dimensional-non-semi-simple-lie-algebra, I just learned many useful comments and papers in the literature.
From what I had learned in the Reference of this post, we can (strictly?) organize the classes of finite dimensional Lie algebra by:
I. semi-simple. (Killing form is non-degenerate)
II. non-semi-simple (Killing form is degenerate):
$\bullet$ non-solvable.
$\bullet$ solvable and nilpotent.
$\bullet$ solvable but not nilpotent.

QUESTION: Here I am simply interested in focusing on: 
"What are the list of 6 dimensional Lie algebra and 8 dimensional Lie algebra, which allow symmetric non-degenerate invariant bilinear forms $\Omega_{ab}$?"
(hopefully the list can be as complete as possible, but a partial list is welcome.)
If there is a list of corresponding $\Omega_{ab}$ metric, it will be the best.

I am mostly interested in non-semi-simple case, and in real Lie algebra more than the complex Lie algebra (of course, if providing examples of complex Lie algebra will also be nice).
eg: So far I know only one example in 6-dimension is the nilpotent Lie algebra $A_{6,3}$ with symmetric nondegenerate $\Omega_{ab}$:
The algebra is 
$$
[e_1,e_2]=e_6,\;\;[e_1,e_3]=e_4,\;\;[e_2,e_3]=e_5,
$$
with other commutators are zeros. One can find the nondegenerate $\Omega_{ab}$ to be:
$$
\Omega_{ab}={\begin{pmatrix} 
q_1 & 0&0&0 &q_2&0 \\
0& q_3 & 0& -q_2&0 &0\\
0& 0& q_4 & 0& 0& q_2\\
0&  -q_2 & 0& 0& 0 &0\\
q_2 &0& 0& 0& 0 &0\\
0& 0& q_2 & 0& 0 &0
\end{pmatrix}}
$$
What are other examples in 6 dimensional Lie algebra and 8 dimensional Lie algebra? (for those semi-simple Lie algebra, I suppose we can use Killing form = $\Omega_{ab}=-{f_{ak}}^l{f_{bl}}^k$. What are the complete lists of semi-simple and non-semi-simple ones of 6 and 8 dimensions?)
Papers/Ref are mostly welcome. (This question is well-motivated by constructing a type of Wess-Zumino-Witten model). Thank you for the concern.
 A: There are several classification lists of solvable and nilpotent quadratic Lie algebras, i.e., having a symmetric, invariant non-degenerate bilinear form. For the classification of nilpotent quadratic Lie algebras of dimension $n\le 7$ over the field of real and complex numbers, see 
Piu P., Goze M., Gruppi e Algebre di Lie, appunti per un seminario, Universita degli studi di Cagliari, Dipartimento di Mathematica, 1991.
Gr. Tsagas and P. Nerantzi: Symmetric invariant non-degenerate bilinear forms on nilpotent Lie algebras - see here.
It turns out, that in dimension $6$ there is just one indecomposable nilpotent quadratic algebra, and a decomposable arising from the $5$-dimensional and $1$-dimensional quadratic algebra. In dimension $8$ there are many (two-step nilpotent) examples, but I think, no complete classification. 
For the classification of solvable ones in dimension $n\le 6$ see
Tien Dat Pham, Anh vu Le, Minh thanh Duong: Solvable quadratic Lie algebras in low dimension.
In general, see double extension construction and work by Medina and Revoy, Favre and Santaroubane, and many others.
