On the sum of an almost Abelian Lie algebra and a Lie algebra finite-dimensional over its center (Q2730519)

The Lie algebras \(L\) over an arbitrary field are considered which can be decomposed into a sum \(L=A+B\) of an almost Abelian subalgebra \(A\) (i.e. containing an Abelian ideal of finite codimension) and a subalgebra \(B\) which is finite dimensional over its center. It is proved that such an algebra is almost solvable (i.e. it contains a solvable ideal of finite codimension). As a corollary, the following statement is proved:NEWLINENEWLINENEWLINEA Lie algebra over an arbitrary field which is a sum of an Abelian subalgebra and an almost Abelian subalgebra is almost solvable.NEWLINENEWLINENEWLINEThe author notes that it is still unknown whether the analogous statement (in case when both summands are almost Abelian) is true for groups.