Decomposition of matrices into commutators of unipotent matrices of index 2

Abstract

Let $\mathbb{C}$ be the complex field. Denote by $\mathrm{SL}_n(\mathbb{C})$ the group of all complex $n\times n$ matrices with determinant $1$. It is proved that every matrix in $\mathrm{SL}_n(\mathbb{C})$ can be decomposed into a product of two commutators of unipotent matrices of index $2$. Moreover, two is the smallest such number.