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You can use the second syntax of Transpose and it should work like the code below, but it is yet to be checked whether each 4-length List as the second argument of Transpose exactly corresponds to your order of indexes or not

n = {a, b};
id4 = TensorProduct[IdentityMatrix[2], n, n];
mat = TensorProduct[n, n, n, n] - (id4 + Transpose[id4, {1, 3, 2, 4}] + 
     Transpose[id4, {1, 4, 3, 2}] + Transpose[id4, {3, 1, 4, 2}] + 
     Transpose[id4, {3, 1, 2, 4}] + Transpose[id4, {3, 4, 1, 2}])/6