On varieties of commuting triples

The problem of approximating triples of commuting n×n matrices by triples of generic matrices is equivalent to that whether the variety C(3,n) of triples of commuting matrices is irreducible. It is known that the variety is irreducible for n⩽7 and reducible for n⩾30. Using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that C(3,8) is irreducible.