An algorithm for associative bilinear forms

We establish the equivalence between the problem of existence of associative bilinear forms and the problem of solvability in commutative power-associative finite-dimensional nil-algebras. We use the tensor product to find sufficient and necessary conditions to assure the existence of associative bilinear forms in an algebra A. The result gives us an algorithm to describe the space of associative bilinear forms for an algebra when its constants of structure γ_i,j,k for i,j,k=1,…,n are known.