We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan-Kac-Vinberg decompositions of a rational vector all lie in a single rational orbit.

Answering a question of H. Petersson, we provide a class of examples of pair of octonion algebras over a ring having isometric norms.

Suppose we are given a finite-dimensional vector space $V$ equipped with an $F$-rational action of a linearly algebraic group $G$, with $F$ a characteristic zero field. We conjecture the following: to each vector $v\in V(F)$ there corresponds a canonical $G(F)$-orbit of semisimple vectors of $V$. In the case of the adjoint action, this orbit is the $G(F)$-orbit of the semisimple part of $v$, so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.

We embed the moduli space $Q$ of 5 points on the projective line $S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the 6-dimensional irreducible module of the symmetric group $S_5$. This module splits with respect to the icosahedral group $A_5$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of $A_5$ on $Q$ to those on the regular icosahedron.