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.

indecomposable. Otherwise you also have the abelian one, as well as the direct product of the indecomposable 5-dimensional one by the 1-dimensional one. $\endgroup$symmetricnon-degenerate invariant bilinear form? $\endgroup$4more comments