Surjective Stability of Dickson-Siegel-Eichler-Roy Elementary Orthogonal Group

Let $R$ be a commutative Noetherian ring in which $2$ is invertible.

We prove that a conjugate of Petrov's odd elementary unitary group is contained in the DSER elementary orthogonal group defined over projective modules.

We also show a sufficient condition regarding the Witt index of the quadratic module with a hyperbolic summand $\mathbb{H}(P)$ which implies the surjective stability of DSER orthogonal ${\rm K}_1$