  We show that for a reductive group $G$ over a field $k$ the
$\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an
invertible element of the Grothendieck-Witt ring $\mathrm{GW}(k)$, settling the
weak form of a conjecture by Fabien Morel. As an application we obtain a
generalized splitting principle which allows one to reduce the structure group
of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.
