In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am surprised to be unable to find any parallel discussion of octonion algebras in the literature, and wonder if someone may know a reference. The lack of associativity is what seems to make it hard to adapt Grothendieck's calculations.

## Is it known (better: proved somewhere citable?) that any two octonion algebras over an artin local ring with algebraically closed residue field (of any characteristic) are necessarily isomorphic?

The non-associativity is very disorienting to me for dealing with the cohomological obstruction. I'd prefer not to reinvent a useful non-associative wheel.

Some remarks to avoid confusion:

Remark 1: An *octonion algebra* over a scheme $S$ is a rank-8 non-degenerate quadratic space $(V,q)$ equipped with a (not necessarily associative) $O_S$-algebra structure admitting 2-sided identity $e \in V(S)$ such that $q(xy) = q(x)q(y)$ (which forces $q(e)=1$). It can be deduced from this that for the perfect bilinear $B_q(x,y) := q(x+y) - q(x)-q(y)$ and the linear form $t = B_q(\cdot,e)$, the operation $x \mapsto x^{\ast} := t(x)-x$ is an anti-involution of the algebra, $x x^{\ast} = x^{\ast} x = q(x)$, $B_q(x,y) = t(x y^{\ast})$, and every algebra automorphism automatically preserves $q$ (hence also $t$ and the conjugation).

Remark 2: There is a recent paper developing a low-degree cohomology/obstruction theory for left alternative algebras (which has a hypothesis of alg. closed ground field of char. 0 that is never used in the main proofs). In that theory, vanishing of ${\rm{H}}^2$ is equivalent to triviality of all deformations *as an alternative algebra*, but it isn't clear that this framework should be helpful adequate for dealing with octonion algebras. It seems unlikely, since in the same paper it is shown that the "left alternative" ${\rm{H}}^2$ for a $2 \times 2$ matrix algebra is nonzero, so that matrix algebra has nontrivial deformations as an alternative algebra (which are irrelevant if one cares about deforming it as an associative algebra).

Remark 3: By computing derivations over a field, we know that the automorphism scheme of an octonion algebra over a field is smooth, even connected semisimple of type ${\rm{G}}_2$. By a trick with Dedekind domains I know that the automorphism scheme of an octonion algebra over a Dedekind domain is actually flat, so (by the theory over fields) it is a semisimple group scheme of type ${\rm{G}}_2$. I do not know what happens for a more general base ring.

The question above is equivalent to asking for smoothness of the Isom-scheme between a pair of octonion algebras over a general base (and it is equivalent to flatness as well, without which it isn't clear how to show the Isom-scheme is a torsor for the automorphism scheme of either of the given octonion algebras). It is also equivalent to asking that every octonion algebra becomes "split" etale-locally on the base, which is a classical fact for fields but I have no idea if it is true over the dual numbers over a field.

For example, in lieu of knowing a positive answer to my question, it isn't clear (to me) that the study of octonion algebras is completely controlled by the study of ${\rm{G}}_2$-bundles.

5more comments