T1 = 1/2 * {{0,1,0,0}, {1,0,0,0},{0,0,0,0},{0,0,0,0}}
T2 = 1/2 * {{0,-I,0,0},{I,0,0,0},{0,0,0,0},{0,0,0,0}}
T3 = 1/2 * {{1,0,0,0}, {0,-1,0,0},{0,0,0,0},{0,0,0,0}}

T4 = 1/2*{{0, 0, 1,0}, {0, 0, 0,0},{1,0, 0, 0},{0,0,0,0}}
T5 = 1/2*{{0, 0, -I,0}, {0, 0,0,0},{I,0,0,0},{0,0,0,0}}

T6 = 1/2*{{0, 0, 0,0}, {0, 0, 1,0}, {0, 1, 0,0},{0,0,0,0}}
T7 = 1/2*{{0, 0, 0,0}, {0, 0, -I,0}, {0, I, 0,0},{0,0,0,0}}
T8 = 1/(2*Sqrt[3]) {{1, 0, 0,0}, {0, 1, 0,0}, {0, 0, -2,0},{0,0,0,0}}

T9 = 1/2 * {{0,0,0,1}, {0,0,0,0},{0,0,0,0},{1,0,0,0}}
T10 = 1/2 * {{0,0,0,-I}, {0,0,0,0},{0,0,0,0},{I,0,0,0}}

T11 = 1/2 * {{0,0,0,0}, {0,0,0,1},{0,0,0,0},{0,1,0,0}}
T12 = 1/2 * {{0,0,0,0},{0,0,0,-I},{0,0,0,0},{0,I,0,0}}

T13 = 1/2 * {{0,0,0,0}, {0,0,0,0},{0,0,0,1},{0,0,1,0}}
T14 = 1/2 * {{0,0,0,0},{0,0,0,0},{0,0,0,-I},{0,0,I,0}}

T15 = 1/(2*Sqrt[6]) {{1, 0, 0,0}, {0, 1, 0,0}, {0, 0, 1,0},{0,0,0,-3}}


E1  = T1 + I*T2
Em1  =T1 - I*T2

E2  = T4 + I*T5
Em2  =T4 - I*T5

E3  = T6 + I*T7
Em3  =T6 - I*T7

E4  = T9 + I*T10
Em4  =T9 - I*T10

E5  = T11 + I*T12
Em5  =T11 - I*T12

E6  = T13 + I*T14
Em6  =T13 - I*T14

w1={T3[[1,1]],T8[[1,1]],T15[[1,1]]}
w2={T3[[2,2]],T8[[2,2]],T15[[2,2]]}
w3={T3[[3,3]],T8[[3,3]],T15[[3,3]]}
w4={T3[[4,4]],T8[[4,4]],T15[[4,4]]}

r1 = {a, b, c};
r1=r1 /. Solve[(T3.E1 - E1.T3) == a*E1, a];
r1=r1 /. Solve[(T8.E1 - E1.T8) == b*E1, b];
r1=r1 /. Solve[(T15.E1 - E1.T15) == c*E1, c];
r1 = Flatten[r1]

r2 = {a, b, c};
r2=r2 /. Solve[(T3.E2 - E2.T3) == a*E2, a];
r2=r2 /. Solve[(T8.E2 - E2.T8) == b*E2, b];
r2=r2 /. Solve[(T15.E2 - E2.T15) == c*E2, c];
r2 = Flatten[r2]

r3 = {a, b, c};
r3=r3 /. Solve[(T3.E3 - E3.T3) == a*E3, a];
r3=r3 /. Solve[(T8.E3 - E3.T8) == b*E3, b];
r3=r3 /. Solve[(T15.E3 - E3.T15) == c*E3, c];
r3 = Flatten[r3]

r4 = {a, b, c};
r4=r4 /. Solve[(T3.E4 - E4.T3) == a*E4, a];
r4=r4 /. Solve[(T8.E4 - E4.T8) == b*E4, b];
r4=r4 /. Solve[(T15.E4 - E4.T15) == c*E4, c];
r4 = Flatten[r4]

r5 = {a, b, c};
r5=r5 /. Solve[(T3.E5 - E5.T3) == a*E5, a];
r5=r5 /. Solve[(T8.E5 - E5.T8) == b*E5, b];
r5=r5 /. Solve[(T15.E5 - E5.T15) == c*E5, c];
r5 = Flatten[r5]

r6 = {a, b, c};
r6=r6 /. Solve[(T3.E6 - E6.T3) == a*E6, a];
r6=r6 /. Solve[(T8.E6 - E6.T8) == b*E6, b];
r6=r6 /. Solve[(T15.E6 - E6.T15) == c*E6, c];
r6 = Flatten[r6]


rm1 = {a, b, c};
rm1=rm1 /. Solve[(T3.Em1 - Em1.T3) == a*Em1, a];
rm1=rm1 /. Solve[(T8.Em1 - Em1.T8) == b*Em1, b];
rm1=rm1 /. Solve[(T15.Em1 - Em1.T15) == c*Em1, c];
rm1 = Flatten[rm1]

rm2 = {a, b, c};
rm2=rm2 /. Solve[(T3.Em2 - Em2.T3) == a*Em2, a];
rm2=rm2 /. Solve[(T8.Em2 - Em2.T8) == b*Em2, b];
rm2=rm2 /. Solve[(T15.Em2 - Em2.T15) == c*Em2, c];
rm2 = Flatten[rm2];

rm3 = {a, b, c};
rm3=rm3 /. Solve[(T3.Em3 - Em3.T3) == a*Em3, a];
rm3=rm3 /. Solve[(T8.Em3 - Em3.T8) == b*Em3, b];
rm3=rm3 /. Solve[(T15.Em3 - Em3.T15) == c*Em3, c];
rm3 = Flatten[rm3]

rm4 = {a, b, c};
rm4=rm4 /. Solve[(T3.Em4 - Em4.T3) == a*Em4, a];
rm4=rm4 /. Solve[(T8.Em4 - Em4.T8) == b*Em4, b];
rm4=rm4 /. Solve[(T15.Em4 - Em4.T15) == c*Em4, c];
rm4 = Flatten[rm4]

rm5 = {a, b, c};
rm5=rm5 /. Solve[(T3.Em5 - Em5.T3) == a*Em5, a];
rm5=rm5 /. Solve[(T8.Em5 - Em5.T8) == b*Em5, b];
rm5=rm5 /. Solve[(T15.Em5 - Em5.T15) == c*Em5, c];
rm5 = Flatten[rm5]

rm6 = {a, b, c};
rm6=rm6 /. Solve[(T3.Em6 - Em6.T3) == a*Em6, a];
rm6=rm6 /. Solve[(T8.Em6 - Em6.T8) == b*Em6, b];
rm6=rm6 /. Solve[(T15.Em6 - Em6.T15) == c*Em6, c];
rm6 = Flatten[rm6]

2*(r1[[1]]*T3 + r1[[2]]*T8 + r1[[3]]*T15) === E1.Em1 - Em1.E1
2*(r2[[1]]*T3 + r2[[2]]*T8 + r2[[3]]*T15) === E2.Em2 - Em2.E2
2*(r3[[1]]*T3 + r3[[2]]*T8 + r3[[3]]*T15) === E3.Em3 - Em3.E3
2*(r4[[1]]*T3 + r4[[2]]*T8 + r4[[3]]*T15) === E4.Em4 - Em4.E4
2*(r5[[1]]*T3 + r5[[2]]*T8 + r5[[3]]*T15) === E5.Em5 - Em5.E5
2*(r6[[1]]*T3 + r6[[2]]*T8 + r6[[3]]*T15) === E6.Em6 - Em6.E6