"My" code is just your code. I didn't change anything that I recall.
L[u_] = Array[L, {3, 3}] + {{u, 0, 0}, {0, u, 0}, {0, 0, u}};
t[\[Epsilon]] = {{1, \[Epsilon]1, \[Epsilon]2}, {-\[Epsilon]1,
1, \[Epsilon]3}, {-\[Epsilon]2, -\[Epsilon]3, 1}};
T[u_] = L[u].t[\[Epsilon]];
B[u_] := T[u][[2, 3]] (T[u][[1, 2]] T[u - v][[2, 3]] -
T[u][[1, 3]] T[u - v][[2, 2]]) -
T[u][[1, 3]] (T[u][[1, 3]] T[u - v][[2, 1]] -
T[u][[1, 1]] T[u - v][[2, 3]]);
L[1, 1] = I/3 {{2, 0, 0}, {0, -1, 0}, {0, 0, -1}};
L[2, 2] = I/3 {{-1, 0, 0}, {0, 2, 0}, {0, 0, -1}};
L[3, 3] = I/3 {{-1, 0, 0}, {0, -1, 0}, {0, 0, 2}};
L[1, 2] = I {{0, 0, 0}, {1, 0, 0}, {0, 0, 0}};
L[1, 3] = I {{0, 0, 0}, {0, 0, 0}, {1, 0, 0}};
L[2, 1] = I {{0, 1, 0}, {0, 0, 0}, {0, 0, 0}};
L[2, 3] = I {{0, 0, 0}, {0, 0, 0}, {0, 1, 0}};
L[3, 1] = I {{0, 0, 1}, {0, 0, 0}, {0, 0, 0}};
L[3, 2] = I {{0, 0, 0}, {0, 0, 1}, {0, 0, 0}};
u = Array[x, {3, 3}];
v = I {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
sol = u /. Solve[Flatten[B[u]] == 0, Flatten[u]];
Length[sol]
(* Out[18]= 432 *)
Here is one of them.
sol[[1]]
(* Out[19]= {{-((2*I)/3), 0, 0}, {0, I/3, 0},
{-((I*(2*\[Epsilon]1*\[Epsilon]2 - \[Epsilon]2^2*\[Epsilon]3 - \[Epsilon]3^3 -
I*\[Epsilon]3*Sqrt[4*\[Epsilon]1^2 - \[Epsilon]2^4 -
2*\[Epsilon]2^2*\[Epsilon]3^2 - \[Epsilon]3^4]))/
(2*(\[Epsilon]1*\[Epsilon]2^2 + \[Epsilon]1*\[Epsilon]3^2))), 0, I/3}} *)