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 Autonne–Takagi factorization (although it was also discovered by Hua and others).
I would like to emphasise that this is not the usual spectral theorem that applies to normal matrices as complex symmetric matrices do not have to be normal.