Cartan subalgebras with Engel decomposition (Q1810233)

Let \(L\) be a Lie algebra over an algebraically closed field \(K\). If char \(K=0\), then it is well known that all Cartan subalgebras of \(L\) have the same dimension (in fact, they are conjugate under the automorphism group of \(L\)). However, when char \(K>0\), this assertion is not valid in general, and is valid only for Cartan subalgebras satisfying certain additional conditions [cf., e.g., \textit{A. A. Premet}, Izv. Akad. Nauk SSSR, Ser. Mat. 50, 788-800 (1986; Zbl 0613.17009)]. In the article under review, the following class of Cartan subalgebras is considered. Let \(H\) be a Cartan subalgebra of \(L\) and \(L=H\bigoplus_{\alpha \neq 0}L_{\alpha}\) the corresponding Cartan decomposition. If for every \(x\in L_\alpha,\) \(\alpha \neq 0\), \(\operatorname {ad}x\) is nilpotent, then \(H\) is called a Cartan subalgebra with Engel decomposition (CSED). The author shows that all CSED's have the same dimension, and a CSED is the centralizer of a regular element (an element \(x\) of \(L\) is regular if the Fitting null part of \(\text{ad }x\) is of minimal dimension). Moreover, it is shown that if \(H_1\) and \(H_2\) are two CSED's, there exists a certain bijective linear transfomation \(\psi\) of \(L\) which maps \(H_1\) onto \(H_2\).