Structure of finitely generated commutative alternative algebras and special Moufang loops. (Q881116)

The following statements are proved: 1) the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank \(n\) is nilpotent of index \(n-1\); 2) the loop of invertible elements of a commutative altenative algebra \(A\) of rank \(n\) with unit 1 is centrally nilpotent of index \(\leq n-1\), the estimate is sharp; 3) any special commutative Moufang loop of rank \(n\) is centrally nilpotent of index \(\leq n-1\), the estimate is sharp (Bruck's theorem for special commutative Moufang loops).