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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. [ABSTRACT FROM AUTHOR]