Decompositions of linear spaces induced by bilinear maps

Let V be an arbitrary linear space and f:V×V→V a bilinear map. We show that, for any choice of basis B of V, the bilinear map f induces on V a decompositionV=⨁j∈JVj as a direct sum of linear subspaces, which is f-orthogonal in the sensef(Vj,Vk)=0 when j≠k, and in such a way that any Vj is strongly f-invariant in the sensef(Vj,V)+f(V,Vj)⊂Vj. We also characterize the f-simplicity of any Vj. Finally, an application to the structure theory of arbitrary algebras is also provided.