On the length of a class of maximal commutative subalgebras

A maximal commutative subalgebra is a substructure in algebra with the greatest commutative property. By studying the lengths of maximal commutative subalgebras, one can more clearly characterize the structure of commutative subalgebras in the full matrix algebra ${\mathrm{M}}_n(\mathbb{F})$. Inspired by \cite[Proposition~4.12]{markova2013}, this paper identifies a class of maximal commutative subalgebras ${\mathcal{B}}_{k,m,l}$ and computes their lengths. Finally, we present two concrete examples to show that it is not a straightforward generalization.