K = 3 and two links with g^2 couplings

Here are the eigenvalues for the constants:
{mub, muf, g2, kappas, kappaa, beta, lambda1, lambda2, lambda3, lambda4, lambda5} = 
{0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0}

K = 3 and two links with g^2 nonzero

In[135]:=
expr = N[generalMatrix[makeham, 3, 2, 0, 1, {0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0},
         none]];

In[137]:=
Sort[Eigenvalues[expr]]

Out[137]=
{-4.163336342344337*^-16, -3.3306690738754696*^-16, -1.1102230246251565*^-16, 0., 
  0.3369424417896301, 0.3369424417896302, 0.33694244178963023, 0.3369424417896303, 
  0.33694244178963034, 0.3369424417896306, 0.3369424417896306, 0.3369424417896306, 
  0.3369424417896306, 0.3369424417896306, 0.3369424417896306, 0.3369424417896307, 
  0.3369424417896309, 0.33694244178963106, 0.3369424417896311, 0.3369424417896313, 
  0.7161972439135288, 0.7161972439135289, 0.716197243913529, 0.7161972439135291, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.804231086598284, 0.804231086598284, 0.804231086598284, 0.804231086598284, 
  0.9060257059985932, 0.9060257059985939, 0.9060257059985939, 0.9060257059985939, 
  0.9060257059985939, 0.9060257059985939, 0.906025705998594, 0.9060257059985941, 
  0.9060257059985942, 0.9060257059985943, 0.9060257059985943, 0.9060257059985943, 
  0.9060257059985943, 0.9060257059985944, 0.9060257059985944, 0.906025705998595, 
  1.4323944878270578, 1.4323944878270578, 1.432394487827058, 1.432394487827058, 
  1.807367433034494, 1.8073674330344944, 1.8073674330344944, 1.8073674330344947, 
  1.807367433034495, 1.807367433034495, 1.8073674330344953, 1.8073674330344953, 
  1.8073674330344953, 1.8073674330344955, 1.8073674330344955, 1.8073674330344955, 
  1.8073674330344955, 1.8073674330344955, 1.8073674330344958, 1.807367433034496}

These are the diagonal matrix elements 

In[138]:=
diag = Table[expr[[i, i]], {i, Length[expr]}]

Out[138]=
{0.596831, 0.596831, 0.596831, 0.596831, 0.95493, 0.95493, 0.95493, 0.95493, \
0.596831, 0.596831, 0.596831, 0.596831, 0.906026, 0.906026, 0.906026, \
0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 1.23828, 1.23828, 1.23828, \
1.23828, 1.23828, 1.23828, 1.23828, 1.23828, 0.906026, 0.906026, 0.906026, \
0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 1.23828, 1.23828, 1.23828, \
1.23828, 1.23828, 1.23828, 1.23828, 1.23828, 0.906026, 0.906026, 0.906026, \
0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 0.906026, \
0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 0.906026, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, \
0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231, 0.804231}

This is a list of the different elements of the matrix

In[136]:=
Union[Flatten[expr]]

Out[136]=
{-0.506428, -0.477465, -0.119366, 0., 0.596831, 0.804231, 0.906026, 0.95493, \
1.23828}