Nilpotent alternative algebras (Q2266114)

Let A be a nilpotent alternative algebra over a field F and \(A^ 1=A\), \(A^{k+1}=A^ kA+AA^ k\) for \(k\geq 1\). \((n_ 1,n_ 2,...,n_ s)\) is called the type of A if \(A^ s\neq 0\) and \(A^{s+1}=0\) and dim \(A^ i/A^{i+1}=n_ i\) \((i=1,2,...,s)\). In this paper, the author proves that if dim \(A\leq 5\), then A is an associative algebra, if dim A\(=6\) and A is not associative then the type of A is (3,2,1) and if dim A\(=7\) and A is not associative then the type of A is one of the followings: (4,2,1), (4,1,2), (3,3,1), (3,1,3), (3,2,2), (3,2,1,1), (3,1,2,1), (3,1,1,2). Finally, the multiplication table of the generators of A which type as above is given.