Abstract

We study commutative algebras satisfying the identity (xx)(xx) - lambda((xx)x)x = 0. It is known that for lambda = 1 and for characteristic not 2,3 or 5, the algebra is a commutative power-associative algebra. These algebras have been widely studied by Albert, Gerstenhaber and Schafer. For lambda = 0, Guzzo and Behn in 2014 proved that commutative algebras of dimension <= 7 satisfying (xx)(xx) = 0 are solvable. We consider the remaining values of lambda. We prove that commutative algebras satisfying (xx)(xx)-lambda((xx)x)x = 0 with lambda not equal 0, 1, and generated by one element are nilpotent of nilindex <= 8 (we assume characteristic of the field not equal 2, 3).