Commutative nilpotent subalgebras with nilpotency index $n-1$ in the algebra of matrices of order $n$

The paper establishes the existence of an element with nilpotency index $n-1$ in the algebra of upper niltriangular matrices $N_n(\mathbb F)$ over a field $\mathbb F$ with at least $n$ elements for all $n\ge5$ and, as a corollary, also in the full matrix algebra $M_n(\mathbb F)$. This result implies an improvement with respect to the basic field of known classification theorems due to D. A. Suprunenko, R. I. Tyschkevich, and I. A. Pavlov for algebras of the class considered.