  We introduce in this note the notion of the category of twisted Chow-Witt
correspondences $CHW(k)$ over a field $k$ of characteristic different from $2$.
Moreover, we show that over an infinite perfect field this category $CHW(k)$
admits a fully faithful embedding into the geometric stable $\A^1$-derived
category $D_{\A^1,gm}(k)$ after taking $\Q$-localization. We also prove a
conjecture of F. Morel about the rational splitting of stable $\A^1$-cohomology
over an essentially smooth scheme $S$ over a field $k$ of $char(k) \neq 2$.
