Determine the F and D matrices for the triplet state.  First start with a 3x3 matrix:

table = Table[0,{3},{3}];

Make the isospin raising operator:

\(tp\  = \ table; 
tp[\([1, 2]\)] = 1; 
tp // MatrixForm\)

And the lowering operator:

(tm = Transpose[tp]) // MatrixForm

And t_x:

(tx = (tp + tm)/2) // MatrixForm

and t_y:

(ty = (tp  -   tm)/2/I) // MatrixForm

Define the commutator and the anticommutator:

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

anticommutator[a_, b_]  =  a . b  +  b . a

Get t_z from commutator relationship:

(tz  =  -I commutator[tx, ty]) // MatrixForm

Check total isospin:

(Isq = tx . tx + ty . ty + tz . tz) // MatrixForm

\(up\  = \ table; 
up[\([2, 3]\)] = 1; 
up // MatrixForm\)

(um = Transpose[up]) // MatrixForm

(ux = (up + um)/2) // MatrixForm

(uy = (up  -   um)/2/I) // MatrixForm

(uz  =  -I commutator[ux, uy]) // MatrixForm

(Usq =  ux . ux + uy . uy + uz . uz) // MatrixForm

\(vp\  = \ table; 
vp[\([1, 3]\)] = 1; 
vp // MatrixForm\)

(vm = Transpose[vp]) // MatrixForm

(vx = (vp + vm)/2) // MatrixForm

(vy = (vp  -   vm)/2/I) // MatrixForm

(vz  = - I commutator[vx, vy]) // MatrixForm

(Vsq =  vx . vx + vy . vy + vz . vz) // MatrixForm

\(\(f1 = tx; 
f2 = ty; 
f3 = tz; 
f4 = vx; 
f5 = vy; 
f6 = ux; 

    f7 = uy; 
Y = 2/3 \((2  uz\  + \ tz)\); 

    \((f8 = Y*Sqrt[3]/2)\) // MatrixForm
\)\)

flist={f1,f2,f3,f4,f5,f6,f7,f8};

F[i_] := flist[[i]];

Table[
        If[commutator[F[i], F[j]] == table, 0, 1], {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]]

d[i_] := 2/3 Sum[ddtable[i,j,k]F[j].F[k],

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

d[1]//MatrixForm

d[2]//MatrixForm

commutator[d[1], F[2]] // MatrixForm

d[3]//MatrixForm

d[4]//MatrixForm

d[5]//MatrixForm

d[6]//MatrixForm

d[7]//MatrixForm

d[8]//MatrixForm

\(\(
Table[
      If[\((anticommutator[F[i], F[j]] - 
              1/3  IdentityMatrix[3]*If[i == j, 1, 0] - 
              Sum[ddtable[i, j, k] F[k], {k, 1, 8}])\) != Table[0, {3}, {3}], 
        1, 0], {i, 1, 8}, {j, 1, 8}]\)\)