Similar to the multiplication of square matrices one can define multiplications for three dimensional matrices, i.e., for the "cubes" of the vector space

where I denotes a finite set of indices and Kis any field. The multiplications shall imitate the matrix multiplication: To obtain the coefficient γxyzof the product (γxyz) — (αxyz)( βxyz),all coefficients axij, ij∈ I, of the horizontal plane with index xof (αxyz)are multiplied with certain coefficients βhgzof the vertical plane with index z of (βxyz)and the results are added: