Symmetric, invariant, non-degenerate bilinear form on a Lie algebra
Bearing in mind the importance of the Killing form, this paper is concerned with the class of regular quadratic Lie algebras \((g,\varphi)\) where \(g\) is a Lie algebra endowed with a symmetric, invariant, nondegenerate bilinear form \(\varphi\). It turns out that such Lie algebras, which are indecomposable and have nontrivial centre, are of the form \((g_ d,\varphi_ d)\) (where \(d\) is an appropriate derivation of \(g\)), and \((g_ d,\varphi_ d)\) is called the extension of \((g,\varphi)\) by \(d\). This enables classification problems to be tackled, and in particular the case when \(d\) is nilpotent. Some interesting open questions are stated, and the paper concludes with the list of all regular quadratic nilpotent Lie algebras of dimension \(\leq 7\).
scientific article; zbMATH DE number 3983438