2-Generated nilpotent algebras and Eggert's conjecture

Let A be a commutative nilpotent finitely-dimensional algebra over a field F of characteristic p>0. A conjecture of Eggert (1971) [4] says that , where A(p) is the subalgebra of A generated by elements ap, a∈A. We show that the conjecture holds if A(p) is at most 2-generated.We give a complete characterization of 2-generated nilpotent commutative algebras in the terms of standard basis with respect to the reverse lexicographical ordering.