The octonions on a bad day

Question

We can define the algebra of quaternions $\mathbb H$ over any field $k$, and depending on the arithmetic of $k$ it is either a division algebra or a matrix algebra.

We can also define the algebra of octonions $\mathbb O$ over any field $k$, and if over $k$ the $8$-ary quadratic form $Q=x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2+x_7^2$ is anisotropic, then $\mathbb O$ is again a division algebra —a non-associative one, but oh well.

What happens to $\mathbb O$ if $k$ is such that $Q$ is isotropic?

The classical structure theory of non-commutative Jordan algebras tells us that $\mathbb O$ is, over any field, a direct product of simple flexible power-associative algebras coming from a rather restricted list: simple commutative Jordan algebras, quasiassociative algebras, and flexible quadratic algebras with nondegenerate norm forms (Shafer's book on non-associative algebra explains all this, which is —I'd say— mostly forgotten nowadays) but I am pretty sure one can be very specific about what comes out in the case of octonions. In other words, one can probably find something playing the role of «matrix algebra» in the statement about quaternions.

N.B. All this is over fields of characteristic zero.

Answer 1

$\DeclareMathOperator\Tr{Tr} $Suppose that $k$ is a field with $\operatorname{char}(k) \neq 2$. Let's agree that an "octonion algebra" over $k$ is an 8-dimensional unital $k$-algebra $A$, endowed with a quadratic form $N: A \rightarrow k$, whose associated bilinear form $T(x,y) = N(x+y) - N(x) - N(y)$ is nondegenerate, and which satisfies $N(xy) = N(x) N(y)$ for all $x,y \in A$.

When $x \in A \setminus k$, the trace $\Tr(x) = T(x,1)$ and norm $N(x)$ are determined by the algebra structure: $x$ is the root of a quadratic polynomial with coefficients $-\Tr(x)$ and $N(x)$. Hence the algebra structure determines the quadratic form $N$ in a convenient way. One can talk about the "isomorphism class of an octonion algebra" while carrying around the quadratic form or not; it doesn't really matter.

It is an old theorem (Jacobson? Albert? I can't recall … check "The Book of Involutions" too) that the isomorphism class of the octonion algebra $k$ is determined by the isomorphism class of the quadratic form $N$. Now, essentially by the Cayley–Dickson doubling process, the norm form $N$ is a Pfister form, i.e., $N$ is isomorphic to $\langle1,-a\rangle \otimes \langle1,-b\rangle \otimes \langle1,-c\rangle$ for some $a,b,c \in k$.

It is a fact about Pfister forms that when they are isotropic, they are split. So if the norm form represents zero, then $N$ is isomorphic to $\langle1,-1\rangle \otimes \langle1,-1\rangle \otimes \langle1,-1\rangle$ and the octonion algebra is isomorphic to the split octonion algebra over $k$.

In this level of generality, the isomorphism classes of octonion algebras over $k$ are classified up to isomorphism by the Galois cohomology $H^3(k, \mu_2)$; you can see such a cohomology class too from the Pfister form perspective. The Pfister form $\langle1,-a\rangle \otimes \langle1,-b\rangle \otimes \langle1,-c\rangle$ depends only on the square-classes of $a, b, c$, giving three classes in $H^1(k, \mu_2)$ whose cup product is the element of $H^3(k, \mu_2)$ classifying the octonion algebra.

Answer 2

@Marty has already answered this question; this answer is to provide additional context. Marty’s answer addresses the case where $k$ is a field of characteristic different from 2. But the results cited actually hold for all fields regardless of the characteristic - a good general reference is

• Springer, Tonny A.; Veldkamp, Ferdinand D., Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-66337-1/hbk). viii, 208 p. (2000). ZBL1087.17001.

Yet more is true. The same results hold even when $k$ is merely a semilocal ring (or even a slightly more general thing called an LG ring), see

• Garibaldi, S.; Petersson, H.P.; Racine, M.L. Albert algebras over commutative rings, New Mathematical Monographs vol. 48. Cambridge (ISBN 9781009426862) 2024. online PDF

The statement “the isomorphism class of the norm form determines the isomorphism class of the octonion algebra” is Theorem 23.5 and the statement “If the norm is isotropic, then the octonion algebra is split” is Theorem 22.18.

However, neither of those statements holds for all commutative rings $k$, even if you assume $k$ contains a field. For counterexamples, see

• Gille, Philippe, Octonion algebras over rings are not determined by their norms, Can. Math. Bull. 57, No. 2, 303-309 (2014). ZBL1358.11126.
• Asok, Aravind; Hoyois, Marc; Wendt, Matthias, Generically split octonion algebras and $\mathbb{A}^1$-homotopy theory, Algebra Number Theory 13, No. 3, 695-747 (2019). ZBL1430.14051. See especially Theorem 4.3.6.