My question is concerning Lemma 2.3 in the following paper , "On the Irreducibility of Commuting Varieties of Nilpotent Matrices "
Specifically, I'm trying to understand the second half of the proof where Basili shows that we can construct a basis that will give us the desired upper triangular matrix. I'm not quite following their notation here. If somebody could explain this in more explicit terms than are used here that would be great.