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.

## Best Answer

The Killing form is a complex quadratic-form and therefore given by complex symmetric (note - not Hermitian!) matrix (with respect to some basis). \begin{align} B(x,y) = x^T A y, \end{align} complex symmetric matrices may be diagonalised by a unitary and it's transpose (note - not conjugate-transpose!). I.e. for any complex symmetric $A$ there exists a unitary $U$ and diagonal matrix $D$, with real, non-negative entries such that \begin{align} D = U A U^T. \end{align} This is known as the Autonneâ€“Takagi factorization (although it was also discovered by Hua and others).

I would like to emphasise that this is

notthe usual spectral theorem that applies to normal matrices as complex symmetric matrices do not have to be normal.