  Let $G_k$ be a connected reductive algebraic group over an algebraically
closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a
quasi-split symmetric subgroup of $G_k$ with respect to an involution
$\theta_k$ of $G_k$. The classification of such involutions is independent of
the characteristic of $k$ (provided not $2$).
  We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the
Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G},
\mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any
algebraically closed field of characteristic $\neq 2$, that is, the geometric
fibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ over
any algebraically closed field $k$ of characteristic $\neq 2$. As a
consequence, we show the coordinate ring of the group $K_k$ is spanned by the
dual $\imath$canonical basis of the corresponding $\imath$quantum group.
  We then construct a quantum Frobenius splitting for the quasi-split
$\imath$quantum group at roots of $1$. This generalizes Lusztig's quantum
Frobenius splitting for quantum groups at roots of $1$. Over a field of
positive characteristic, our quantum Frobenius splitting induces a Frobenius
splitting of the algebraic group $K_k$.
  Finally, we construct Frobenius splittings of the flag variety $G_k / B_k$
that compatibly split certain $K_k$-orbit closures over positive
characteristics. We deduce cohomological vanishings of line bundles as well as
normalities. Results apply to characteristic $0$ as well, thanks to the
existence of the scheme $\mathbf{G}^\imath$. Our construction of splittings is
based on the quantum Frobenius splitting of the corresponding $\imath$quantum
group.
