Horrocks' theorem for odd orthogonal groups

Abstract

We prove Horrocks' theorem for the odd elementary orthogonal group, which gives a decomposition of an orthogonal matrix with entries from a polynomial ring $R\left[X\right]$, over a commutative ring R in which 2 is invertible, as a product of an orthogonal matrix with entries in R and an elementary orthogonal matrix with entries from $R\left[X\right]$.