We prove that the structure group of any Albert algebra over an arbitrary field is R-trivial. This implies the Tits–Weiss conjecture for Albert algebras and the Kneser–Tits conjecture for isotropic groups of type $\mathrm {E}_{7,1}^{78}, \mathrm {E}_{8,2}^{78}$. As a further corollary, we show that some standard conjectures on the groups of R-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type $^1\mathrm {E}_6$.

Let $K$ be a two-dimensional global field of characteristic $\neq 2$ and let $V$ be a divisorial set of places of $K$. We show that for a given $n\geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G=\operatorname{Spin}_{n}(q)$ of nondegenerate $n$-dimensional quadratic forms over $K$ that have good reduction at all $v\in V$ is finite. This result yields some other finiteness properties, such as the finiteness of the genus $\mathbf{gen}_{K}(G)$ and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups $H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$ for $i\geqslant 1$ established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type $\mathsf{G}_{2}$.

We study the Chow group of zero-dimensional cycles for projective homogeneous varieties of semisimple algebraic groups. We show that in many cases this group has no torsion.

We give a short proof of Totaro's theorem that every ${{E}_{8}}$ -torsor over a field $k$ becomes trivial over a finite separable extension of $k$ of degree dividing $d\left( {{E}_{g}} \right)={{2}^{6}}{{3}^{2}}5$ .