Lie algebras in which every soluble subalgebra is either Abelian or almost-Abelian (Q2277566)

Let \({\mathfrak C}^*\) denote the class of Lie algebras L in which every subalgebra of a nilpotent subalgebra H of L is an ideal in the idealizer of H in L. Call a Lie algebra L an (A)-algebra if for x,y\(\in L\) such that \([x,y,y]=0\) we have \([x,y]=0\). The author introduces the classes \({\mathfrak C}^*_ 0\) and \({\mathfrak C}^{(*)}\), Lie algebras in which every soluble subalgebra is abelian, and Lie algebras in which any pair of elements x,y such that \([x,y,y]\in <y>\) satisfies \([x,y]\in <y>\), respectively. The classes \({\mathfrak C}^*_ 0\), \({\mathfrak C}^{(*)}\) and (A)-algebras are characterized, locally finite classes of these are investigated and it is proved that \(L{\mathfrak F}\cap (A)=L{\mathfrak F}\cap {\mathfrak I}_ 0(wsi)^ s,\quad L{\mathfrak F}\cap {\mathfrak C}^{(*)}=L{\mathfrak F}\cap {\mathfrak I}(wsi)^ s,\) and \(L{\mathfrak F}\cap {\mathfrak C}^*_ 0=L{\mathfrak F}\cap {\mathfrak I}(si)^ s.\) It is shown that if L is a locally finite \({\mathfrak C}^*\)-algebra over a field of characteristic zero, then L is a locally finite (A)-algebra, an almost-abelian Lie algebra or a three-dimensional split Lie algebra. Examples are given to show that (A)\(\nleq {\mathfrak I}\), (A)\(\cup {\mathfrak A}_ 0<{\mathfrak C}^{(*)}\), and \({\mathfrak C}^{(*)}<{\mathfrak C}(wasc)\).