Generalised homotopy and commutativity principle

In this paper, we study the action of special $n\times n$ linear (resp. symplectic) matrices which are homotopic to identity on the right invertible $n\times m$ matrices. We also prove that the commutator subgroup of ${\mathrm{O}}_{2n}(R[X])$ is two stably elementary orthogonal for a local ring $R$ with $\frac{1}{2}\in R$ and $n\ge 3.$