Commutators of skew-involutions

Abstract

Let $\mathrm{SL}_n(\mathbb{F})$ be the group of all $n\times n$ matrices over a field $\mathbb{F}$ with determinant $1$. Denote by $I$ ($I_n$) the ($n\times n$) identity matrix. A matrix $A$ is called skew-involution if $A^2=-I$. It is proved that every matrix in $\mathrm{SL}_{2n}(\mathbb{F})$ is a product of at most three commutators of skew-involutions if $\mathbb{F}\ne\mathbb{Z}_3$ and $\mathrm{SL}_{2n}(\mathbb{F})\ne\mathrm{SL}_{2}(\mathbb{Z}_2)$, and at most four commutators of skew-involutions if $\mathbb{F}=\mathbb{Z}_3$ and $n>1$. Every complex symplectic matrix is a product of two commutators of complex symplectic skew-involutions, and every real symplectic matrix is a product of not more than four commutators of real symplectic skew-involutions.