The Chow--Witt rings of the classifying space of quadratically oriented bundles

In this paper we compute the Chow–Witt rings of the classifying space ${\mathrm{B}\mathrm{S}\mathrm{L}}_n^c$ of quadratically oriented vector bundles of rank $n$. We also discuss the corresponding quadratically-oriented cobordism spectrum ${\mathrm{M}\mathrm{S}\mathrm{L}}^c$ and show that it is equivalent to $\mathrm{M}\mathrm{S}\mathrm{L}$ after inverting $\eta$.