multiplication table is described in the Multiplication table section, but it is not unique. Unlike three dimensions, there are many tables because every pair of unit vectors is perpendicular to five other unit vectors, allowing many choices for each cross product. Once we have established a multiplication table, it is then applied to general vectors x and y by expressing x and y in terms of the basis and expanding x × y through bilinearity. Using e1 to e7 for the basis vectors a different multiplication table from the one in the Introduction, leading to a different cross product, is given with anticommutativity by More compactly this rule can be written as with i = 1...7 modulo 7 and the indices i, i + 1 and i + 3 allowed to permute evenly. Together with anticommutativity this generates the product. This rule directly produces the two diagonals immediately adjacent to the diagonal of zeros in the table. Also, from an identity in the subsection on consequences, which produces diagonals further out, and so on. The ej component of cross product x × y is given by selecting all occurrences of ej in the table and collecting the corresponding components of x from the left column and of y from the top row. The result is: As the cross product is bilinear the operator x×– can be written as a matrix, which takes the form The cross product is then given by Different multiplication tables Two different multiplication tables have been used in this article, and there are more. These multiplication tables are characterized by the Fano plane, and these are shown in the figure for the two tables used here: at top, the one described by Sabinin, Sbitneva, and Shestakov, and at bottom that described by Lounesto. The numbers under the Fano diagrams (the set of lines in the diagram) indicate a set of indices for seven independent products in each case, interpreted as ijk → ei × ej = ek. The multiplication table is recovered from the Fano diagram by following either the straight line connecting any three points, or the circle in the center, with a sign as given by the arrows. For example, the first row of multiplications resulting in e1 in the above listing is obtained by following the three paths connected to e1 in the lower Fano diagram: the circular path e2 × e4, the