Nilpotency of the Engel algebras (Q1405700)

Let \(G\) be an anticommutative algebra over a field \(K\) and \(A(G)\) be the associative algebra generated by all operators of right multiplication. \(G\) is called an Engel algebra if for any \(x\in G\) the operator \(R_x: g\to g\circ x\), \(g\in G\) is nilpotent. The authors prove the following main results. (1) If \(G\) is an Engel algebra and \(\dim_KG\leq 4\), then \(G\) is nilpotent. (2) For any \(n\geq 5\), an \(n\)-dimensional Engel algebra exists which is not nilpotent. (3) For any algebra \(G\) with basis \(e_1, e_2, ..., e_n\), if any monomial in generators \(R_{e_i}\) \((i=1,2,..., n)\) of \(A(G)\) whose length does not exceed \(n\) is nilpotent, then the algebra \(G\) is nilpotent. Notice that the proof of the last fact uses the results of \textit{V. A. Ufnarovskii} [Mat. Sb., N. Ser. 128, No. 1, 124--132 (1985; Zbl 0598.15002)].