Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.
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