If  $\mathrm{\ρ}\: \mathrm{\𝔤}\mathit{ }\mathit{\to }\mathit{ }\mathrm{gl}\left(V\right)$ is a representation of a solvable Lie algebra $\mathrm{\𝔤}$, 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  $\mathrm{\ρ}$$\left(x\right)\left(Y\right)\= {\mathrm{\λ}}_{x}Y$for all $x \in \mathrm{\𝔤}$.  The eigenvector $Y$ is called a simultaneous eigenvector for the representation .  The vector $Y$may be complex and will, in general, not be unique.  

 Let $\left\{e_1\, e_2\, ..\. \, e_n\right\}$ be the given basis for $𝔤$.  The program RepresentationEigenvector($\mathbf{\ρ}$) returns  the list of eigenvalues $\left[{\mathrm{\λ}}_{e_1}\, {\mathrm{\λ}}_{e_2}\, ..\. \, {\mathrm{\λ}}_{e_n}\right]$ and the simultaneous eigenvector $Y$.

The program RepresentationEigenvector($\mathbf{\ρ}$) works as follows. First a change of basis is made using the program AscendingIdealsBasis so that in the new basis $\left\{f_{1 }\, f_2\, ..\. \, f_n\right\}$the subalgebra spanned by $\left\{f_{1 }\, f_2\, ..\. \, f_k\right\}$ is an ideal in the subalgebra $\left\{f_{1 }\, f_2\, ..\. \, f_{k\+1}\right\}$. Then an eigenspace $E_1\subset V$for the linear transformation $\mathrm{\ρ}\left(f_1\right)$is found. This space is invariant under $\mathrm{\rho }\left(f_2\right)$ so the program next finds a subspace $E_2 \subset E_1$ which is a eigenspace for  $\mathrm{\rho }\left(f_1\right)$and so on.

with(DifferentialGeometry): with(LieAlgebras): with(Library):

L1 := LieAlgebraData([[x2, x4] = x1, [x3 ,x4] = x3], [x1, x2, x3, x4], Alg1);

$\mathrm{L1}:=\left[\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\right]$

DGsetup(L1);

$\mathrm{Lie\; algebra:\; Alg1}$

DGsetup([x1, x2, x3, x4], V);

$\mathrm{frame\; name:\; V}$

M1 := [Matrix([[0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, - 1], [0, 0, 0, 0]]), Matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]), Matrix([[- 1, 0, 0, 0], [0, 0, - 1, 0], [0, 0, 0, 0], [0, 0, 0, 0]])]:

rho1 := Representation(Alg1, V, M1);

RepresentationEigenvector(rho1);

$\left[0\,0\,0\,-1\right]\,\left[\mathrm{D\_x1}\right]$

seq(ApplyRepresentation(rho1, v, D_x1), v = [e1, e2, e3, e4]);

$0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,-\mathrm{D\_x1}$

Invariants(rho1);

$\left[\mathrm{D\_x2}\right]$

P := Matrix([[1, 0, 1, 0], [0, 1, - 1, 0], [2, 0, 1, - 2], [1, 2, 0, 1]]);

rho2 := ChangeRepresentationBasis(rho1, P, "Range", V);

y := RepresentationEigenvector(rho2);

$y:=\left[0\,0\,0\,0\right]\,\left[\mathrm{D\_x1}-\frac{3}{4}\mathrm{D\_x2}-\mathrm{D\_x3}\+\frac{1}{2}\mathrm{D\_x4}\right]$

seq(ApplyRepresentation(rho2, v, y[2][1]), v = [e1, e2, e3, e4]);

$0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,0\mathrm{D\_x1}$

L := Retrieve("Winternitz", 1, [3, 7], Alg2);

$L:=\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{\_a}\mathrm{e1}-\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{\_a}\mathrm{e2}\+\mathrm{e1}\right]$

DGsetup(L):

DGsetup([x1, x2, x3], V2):

rho3 := Representation(Alg2, V2, Adjoint(Alg2));

y := RepresentationEigenvector(rho3);

$y:=\left[0\,0\,-\mathrm{\_a}\+\mathrm{I}\right]\,\left[\mathrm{D\_x1}-\mathrm{I}\mathrm{D\_x2}\right]$

seq(ApplyRepresentation(rho3, v, y[2][1]), v = [e1, e2, e3]);

$0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,-\left(\mathrm{\_a}-\mathrm{I}\right)\mathrm{D\_x1}\+\left(1+\mathrm{I}\mathrm{\_a}\right)\mathrm{D\_x2}$

ChangeFrame(Alg1):

B := evalDG([e1 - e3 + 2*e4, e3, e2 + 2*e4, e1 - e3 + e4]);

$B:=\left[\mathrm{e1}-\mathrm{e3}\+2\mathrm{e4}\,\mathrm{e3}\,\mathrm{e2}\+2\mathrm{e4}\,\mathrm{e1}-\mathrm{e3}\+\mathrm{e4}\right]$

L4 := LieAlgebraData(B, Alg4);

$\mathrm{L4}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-2\mathrm{e2}\,\left[\mathrm{e1}\,\mathrm{e3}\right]\=2\mathrm{e1}-4\mathrm{e2}-4\mathrm{e4}\,\left[\mathrm{e1}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=2\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=-\mathrm{e1}\+3\mathrm{e2}\+2\mathrm{e4}\right]$

DGsetup(L4, [f], [alpha]);

$\mathrm{Lie\; algebra:\; Alg4}$

rho4 := ChangeRepresentationBasis(rho2, B, Alg4);

infolevel[RepresentationEigenvector] := 3:

y := RepresentationEigenvector(rho4);

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]

$y:=\left[0\,0\,0\,0\right]\,\left[\mathrm{D\_x1}-\frac{3}{4}\mathrm{D\_x2}-\mathrm{D\_x3}\+\frac{1}{2}\mathrm{D\_x4}\right]$

seq(ApplyRepresentation(rho4, f||i, y[2][1]) &minus (y[1][i] &mult y[2][1]), i = 1..4);

$0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,0\mathrm{D\_x1}\,0\mathrm{D\_x1}$