Fine decompositions of algebraic systems induced by bases

We consider algebraic systems with several products, including unary products, G (as examples we can take linear spaces, algebras, superalgebras, hom-algebras, triple systems, hom-triple systems, Poisson algebras, Bol algebras, n-algebras, etc.) We show that any basis of G gives rise to a decomposition of G as a direct sum of indecomposable well-described ideals (fine decomposition). The simplicity of the components in this decomposition is also characterized. There are as many non-isomorphic fine decompositions as orbits in a determined action.