Define Commutator:

commutator[a_, b_]  =  a . b  -  b . a

Matrix representation for the 6 representation of SU(3).   Make the following definition of states:

State     Iz     Strangesness Uz   Vz
1           +1           0             0      +1
2             0           0             +1/2 +1/2
3           -1            0            +1     0
4           +1/2       -1           -1/2   0
5           -1/2        -1            0      -1/2
6            0           -2           -1      -1

Create the Isospin raising operator:

MatrixForm[
      iplus = {{0, Sqrt[2], 0, 0, 0, 0}, {0, 0, Sqrt[2], 0, 0, 0}, {0, 0, 0, 
            0, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 
            0}}]

MatrixForm[iminus = Transpose[iplus]]

MatrixForm[f[1]  =  (iplus + iminus)/2]

MatrixForm[f[2]  =  (iplus - iminus)/2/I]

MatrixForm[f[3]  = -I* commutator[f[1], f[2]]]

vplus = Table[0, {6}, {6}];
vplus[[1, 4]] = Sqrt[2];
vplus[[4, 6]] = Sqrt[2];
vplus[[2, 5]] = 1;
MatrixForm[vplus]

MatrixForm[vminus = Transpose[vplus]]

MatrixForm[f[4] = (vplus + vminus)/2]

MatrixForm[f[5] = (vplus - vminus)/2/I]

uplus = Table[0, {6}, {6}];
uplus[[3, 5]] = Sqrt[2];
uplus[[5, 6]] = Sqrt[2];
uplus[[2, 4]] = 1;
MatrixForm[uplus]

MatrixForm[uminus = Transpose[uplus]]

MatrixForm[f[6] = (uplus + uminus)/2]

MatrixForm[f[7] = (uplus - uminus)/2/I]

MatrixForm[
      Y = (commutator[uplus, uminus] + commutator[vplus, vminus])/
          3]

MatrixForm[f[8] = Sqrt[3]/2 Y]

Define the structure functions for the commutators:

fftable[i_, j_, k_] = 0

\(allff[i_, j_, k_, a_] := 
      Module[{}, fftable[i, j, k] = a; 
		fftable[j, i, k] = \(-a\); \ 
        fftable[j, k, i] = a; fftable[k, j, i] = \(-a\); 
        fftable[k, i, j] = a; \ fftable[i, k, j] = \(-a\);]\)

allff[1, 2, 3, 1]

allff[1, 4, 7, 1/2]

allff[1, 5, 6, -1/2]

allff[2, 4, 6, 1/2]

allff[2, 5, 7, 1/2]

allff[3, 4, 5, 1/2]

allff[3, 6, 7, -1/2]

allff[4, 5, 8, Sqrt[3]/2]

allff[6, 7, 8, Sqrt[3]/2]

Check that the commutators are right (table will give a 1 if commutator is incorrect):

Table[
        If[Simplify[
              commutator[f[4], f[8]] - 
                I Sum[fftable[4, 8, k] f[k], {k, 1, 8}]] ≠  
            Table[0, {6}, {6}], 1, 0], {i, 1, 8}, {j, 1, 8}] // 
      TableForm

ddtable[i_,j_,k_] = 0

alldd[i_,j_,k_,a_] := Module[{},ddtable[i,j,k]=a;

	ddtable[j,i,k]=a; ddtable[j,k,i]=a; ddtable[k,j,i]=a;

	ddtable[k,i,j]=a;ddtable[i,k,j]=a;]

alldd[1,1,8,1/Sqrt[3]]

alldd[1,4,6,1/2]

alldd[1,5,7,1/2]

alldd[2,2,8,1/Sqrt[3]]

alldd[2,4,7,-1/2]

alldd[2,5,6, 1/2]

alldd[3,3,8,1/Sqrt[3]]

alldd[3,4,4,1/2]

alldd[3,5,5,1/2]

alldd[3,6,6,-1/2]

alldd[3,7,7,-1/2]

alldd[4,4,8, -1/2/Sqrt[3]]

alldd[5,5,8, -1/2/Sqrt[3]]

alldd[6,6,8, -1/2/Sqrt[3]]

alldd[7,7,8, -1/2/Sqrt[3]]

alldd[8,8,8, -1/Sqrt[3]]

Do[
d[i] = 2/3 Sum[ddtable[i,j,k]f[j].f[k],

	{j,1,8},{k,1,8}],{i,1,8}]

d[3]

\(\(\(
\)\(Table[
        If[commutator[d[i], f[j]] - 
              I\ Sum[fftable[i, j, k]\ d[k], {k, 1, 8}] != 
            Table[0, {6}, {6}], 1, 0], {i, 1, 8}, {j, 1, 8}] // 
      TableForm\)\)\)

Table[a /. Solve[d[i] ==  a f[i], a], {i, 1, 8}]