Two generator subalgebras of Lie algebras

In [ Thompson, J., 1968, Non- solvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74, 383 - 437.], Thompson showed that a finite group G is solvable if and only if every two- generated subgroup is solvable ( Corollary 2, p. 388). Recently, Grunevald et al. [ Grunewald et al., 2000, Two- variable identities in groups and Lie algebras. Rossiiskaya Akademiya Nauk POMI, 272, 161 - 176; 2003. Journal of Mathematical Sciences ( New York), 116, 2972 - 2981.] have shown that the analogue holds for finite- dimensional Lie algebras over infinite fields of characteristic greater than 5. It is a natural question to ask to what extent the two- generated subalgebras determine the structure of the algebra. It is to this question that this article is addressed. Here, we consider the classes of strongly- solvable and of supersolvable Lie algebras, and the property of triangulability.