If rho: g -> gl(V) is a representation of a solvable Lie algebra g, then a fundamental theorem due to Lie (see, for example, Fulton and Harris, page 125 or Varadarajan, page 200) asserts that there is a vector y in V such that rho(x)(y) = lamba_x y for all x in g.  The eigenvector y may be complex and will, in general, not be unique.  The program RepresentationEigenvector(rho) returns one such vector.

The program RepresentationEigenvector(rho) works as follows: First a change of basis is made using the program AscendingIdealsBasis so that in the new basis [f_1, f_2, ..., f_n] the vectors [f_1, f_2, ..., f_k] form an ideal in [f_1, f_2, ..., f_(k + 1)].  Then an eigenspace E1 for rho(f1) is found.  This space is invariant under rho(f2) so the program next finds a subspace E2 in E1 which is an eigenspace for rho(f2) and so on.

The common eigenspace for the first 1 vectors is [-D_x1+D_x4, D_x3, -2*D_x1+D_x2]

The common eigenspace for the first 2 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]
The common eigenspace for the first 3 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]
The common eigenspace for the first 4 vectors is [-4/3*D_x1+D_x2+4/3*D_x3-2/3*D_x4]