A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and $N(x)\ne 0$ for every non-zero elements $x,y\in A$.

Wikipedia writes that two division octonion algebras $(A,N)$ and $(A',N')$ over a field $F$ are $F$-isomorphic if and only if their norms $N$, $N'$ are $F$-isomorphic. This reduces the problem of isomorphic classification of division octonion algebras to the problem of classification of their norms. Is there any additional information on such a classification?

Question. What is the number of non-isomorphic division octonion algebras over a given field? Can it be greater than one? Can it be infinite?

Because finite or algebraically closed fields admit no division octonion algebras, see the lecture notes of Konrad Voelkel.