Transitive action of Lie algebras

The action of a set S of linear operators on a vector space V is, by definition, k-fold transitive if given linearly independent vectors {x1,x2,…,xk} and arbitrary vectors {y1,y2,…,yk}, there is a member A of S with Axi=yi for all i. It is shown that if the action of a Lie algebra of complex matrices is two-fold transitive, then it is either gln(C) or, if n>2, the Lie subalgebra sln(C). Transitive action is not sufficient to yield this conclusion. Infinite-dimensional analogues are also considered.