  A Norton algebra is an eigenspace of a distance regular graph endowed with a
commutative nonassociative product called the Norton product, which is defined
as the projection of the entrywise product onto this eigenspace. The Norton
algebras are useful in finite group theory as they have interesting
automorphism groups. We provide a precise quantitative measurement for the
nonassociativity of the Norton product on the eigenspace of the second largest
eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual
polar graphs, based on the formulas for this product established in previous
work of Levstein, Maldonado and Penazzi. Our result shows that this product is
as nonassociative as possible except for two cases, one being the trivial
vanishing case while the other having connections with the integer sequence
A000975 on OEIS and the so-called double minus operation studied recently by
Huang, Mickey, and Xu.
