Expressing matrices in $\mathrm{SL}_{n}(F)$ as products of commutators of unipotent matrices

Abstract

This paper aims to show that for two positive integers $n \ge k$, every nonscalar matrix in the special linear group of degree $n$ over a field can be written as a product of a maximum of two commutators of unipotent matrices of index $k$. This fact also holds for scalar matrices over a quadratically closed field. Using GAP, some examples are provided to highlight the significance of the field's cardinality and to show that the assumption of quadratically closed fields is essential.