Toward a definition of a Freudenthal trilinear operation (Q2639147)

It is known that a so called Freudenthal trilinear system (FT-system) is a linear space with a bilinear antisymmetric form \(<x,y>\) and a trilinear operation \(\{\) xyz\(\}\) which satisfy some conditions. A weakening of some of these conditions, in particular admitting the degeneration of the form \(<x,y>\), leads to the notion of generalized FT-system. It is proved that the Lie algebras which correspond to the generalized FT-systems are all simple Lie algebras (finite-dimensional and infinite-dimensional Cartan algebras) apart from the infinite-dimensional Lie algebra \(H_ n\). The generalized FT-systems which are not FT-systems in the usual sense are described.