A note on products of commutators of quadratic matrices

Let F be a Pythagorean field of characteristic different from 2 and R be a commutative local ring containing F as a subfield with residue field \(\overline{R}\) $\overline{R}$ . Suppose \(n>1\) $n>1$ is a natural number, \(A=(a_{ij})\in \textrm{SL}_n(R)\) $A=(a_{\mathit{ij}})\in {\text{SL}}_n(R)$ , and q(x) is a quadratic polynomial in F[x] that has a solution in R. The main goal of this paper is to prove that if \(A\in \textrm{SL}_n(R)\) $A\in {\text{SL}}_n(R)$ and \(\overline{A}=(\overline{a_{ij}})\) $\overline{A}=(\overline{a_{\mathit{ij}}})$ is nonscalar, then A is the product of at most two commutators of matrices that are roots of \(q(x)=0\) $q(x)=0$ .