Reductivity of the Lie algebra of a bilinear form

الأصل · EN

Let f:V× V→ F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a a field F, and let L(f) be the subalgebra of (V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to (n) for some n.