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This is an old paper put here for archeological purposes. It is proved that afinite-dimensional Lie algebra over a field of characteristic p>5, that can bewritten as a vector space (not necessarily direct) sum of two nilpotentsubalgebras, is solvable. The same result (but covering also the cases of lowcharacteristics) was established independently by V. Panyukov (Russ. Math.Surv. 45 (1990), N4, 181-182), and the homological methods utilized in theproof were developed later in arXiv:math/0204004. Many inaccuracies in theEnglish translation are corrected, otherwise the text is identical to thepublished version.

A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable