Finite groups generated by 3-transpositions have been introduced and investigated in depth by \textit{B. Fischer} [Invent. Math. 13, 232-246 (1971; Zbl 0232.20040)]. Motivated by this remarkable result, the author attempts to study Lie algebras \(L\) over a field \(\Phi\) with a monomial basis \({\mathcal D}\) (that is, \(\forall a,b \in {\mathcal D}\), \(\exists \alpha \in \Phi\), \(\exists c \in {\mathcal D}\), \(ab = \alpha c)\) satisfying the following condition \((*)\): every pair of elements of \({\mathcal D}\) generates an abelian or 3-dimensional simple Lie subalgebra in \(L\). The subalgebras (ideals) in \(L\) generated by subsets of \({\mathcal D}\) are called \({\mathcal D}\)- subalgebras \(({\mathcal D}\)-ideals). \(L\) is connected if \(L\) has no nontrivial \({\mathcal D}\)-ideals. The rank of \(L\) is the minimum number of elements in \({\mathcal D}\) generating \(L\). First the author shows that every connected Lie algebra of rank 3 satisfying \((*)\) is either a 6-dimensional Lie algebra of type \(2A_ 1\), or one of the 7-dimensional simple Lie algebras listed in Theorem 2.1 (and \(\text{char} \Phi = 2,3)\). Further, he studies connected Lie algebras satisfying \((*)\) and the additional condition \((**)\): \(L\) contains no 7-dimensional simple \({\mathcal D}\)-subalgebras. Theorem A associates to such an algebra \(L\) a certain group \(G(L)\) generated by a conjugacy class of 3-transpositions. Theorem B describes all such Lie algebras under the assumption that \(G(L)\) is isomorphic to the symmetric group \(\Sigma_ \Omega\) of a linearly ordered set \(\Omega\), and the field \(\Phi\) contains all \(\sqrt x\), \(x \in \Phi\). It turns out that, in this situation, all such algebras are isomorphic to each other (and to the Lie algebra of type \(D_ n\) if \(| \Omega | = 2n\), the Lie algebra \(B_ n\) if \(| \Omega | = 2n + 1)\). It should be remarked that Lie algebras with \((*)\), \((**)\) have also been studied by \textit{A. V. Alekseevskij} [Funct. Anal. Appl. 8(1974), 277-279 (1975); translation from Funkts. Anal. Prilozh. 8, 1-4 (1974; Zbl 0307.22008)] and \textit{A. V. Borovik} [Algebra Logic 28, 248-254 (1989); translation from Algebra Logika 28, 382-392 (1989; Zbl 0704.17002)], implicitly, in the context of Jordan subgroups. Thus, it has been observed that the class of Lie algebras with \((*)\), \((**)\) includes Lie algebras of type \(C_{2^ n}\) and \(D_{2^ n}\).