Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(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 𝔰𝔩(n) for some n.