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 ${\rm BSL}_n^c$ of quadratically oriented vector bundles of rank $n$. We also discuss the corresponding quadratically-oriented cobordism spectrum ${\rm MSL}^c$ and show that it is equivalent to $\rm{MSL}$ after inverting $\eta$.