  The paper provides computations of the first non-vanishing
$\mathbb{A}^1$-homotopy sheaves of the orthogonal Stiefel varieties which are
relevant for the unstable isometry classification of quadratic forms over
smooth affine schemes over perfect fields of characteristic $\neq 2$. Together
with the $\mathbb{A}^1$-representability for quadratic forms, this provides the
first obstructions for rationally trivial quadratic forms to split off a
hyperbolic plane. For even-rank quadratic forms, this first obstruction is a
refinement of the Euler class of Edidin and Graham. A couple of consequences
are discussed, such as improved splitting results over algebraically closed
base fields as well as examples where the obstructions are nontrivial.

Algebraic Geometry