Die Summe einer abelschen und einer nilpotenten Lie-Algebra ist auflösbar. (The sum of an abelian and a nilpotent Lie algebra is solvable) (Q1821185)

It is known that a finite group is solvable if it can be written as the product of two nilpotent subgroups. For finite dimensional Lie algebras over a field of characteristic 0, the analogous result is valid [\textit{M. Goto}, J. Sci. Hiroshima Univ., Ser. A 1 26, 1-2 (1962; Zbl 0142.276)]. It is valid at characteristic \(p>0\), if one of the subalgebras is abelian of dimension less than \(p^ 2-p\) [\textit{A. I. Kostrikin}, Vestn. Mosk. Univ., Ser. I. 1982, No.2, 5-8 (1982; Zbl 0492.17006)]. The present paper shows that the condition on the dimension is not necessary for \(p\neq 2\).