*I posted this on math.stackexchange to no avail, so I hope it's appropriate to post here despite that it might not be research-level. I expect the answer to this is well-known to people studying non-associative algebras, but I cannot find it in my references and the more thorough literature on the topic is expensive! On a similar note, I would appreciate a recommendation for a reference covering octonion algebras over number fields and the Cayley-Dickson construction, in more generality. I don't mind expensive if it gives a good thorough treatment.*


Let $K$
be a number field and let $\mathcal{B}=\Big(\frac{a,b}{K}\Big)$
be a quaternion $K$-algebra.
Then its norm is the Pfister form $\langle\langle a,b\rangle\rangle$
over $K$.
Apply the Cayley Dickson construction to $\mathcal{B}$,
yielding an octonion $K$-algebra $\mathcal{C}$.


What is the Pfister form of the norm of $\mathcal{C}$?
Is it $\langle\langle a,b,ab\rangle\rangle$?


Does the isomorphism class of $\mathcal{B}$
(i.e. different choices for $a,b$ preserving the ramification set of $\mathcal{B}$)
determine the isomorphism class of $\mathcal{C}$?
Equivalently, does the isometry class of the quadratic form $\langle\langle a,b\rangle\rangle$
determine the isometry class of the Pfister form of $\mathcal{C}$ as a quadratic form?


If it does, it makes me wonder about the octonion $K$-algebras with norm the Pfister form of $\langle\langle a,b,c\rangle\rangle$, non-isomorphic to $\mathcal{C}$. It would seem that these also contain $\mathcal{B}$ as a quaternion subalgebra,
and that some variations of the Cayley-Dickson construction would take you from $\mathcal{B}$ to these. How does this work?



Does the Cayley-Dickson construction preserve isomorphism of quaternion algebras?
Suppose $K$ is a field, and $B$ your quaternion algebra $K$. The octonion algebra $C$ made from $B$ using the Cayley-Dickson construction depends on an auxiliary choice of an element $c \in K^{\times}$. Namely, $C$ is the set of pairs $(u,v)$ with $u,v \in B$ with addition 


$(u\_1,v\_1) + (u\_2, v\_2) = (u\_1 + u\_2, v\_1 + v\_2)$


and multiplication 


$(u\_1,v\_1)(u\_2,v\_2) = (u\_1u\_2 + c\overline{v\_2}v\_1, v\_2u\_1 +v\_1\overline{u\_2}).$


Here $x \mapsto \overline{x}$ is the involution on the quaternion aglebra $B$. The involution on $C$ is then $(u,v) \mapsto (\overline{u},-v)$. With this multiplication and involution, one computes that the norm on $C$ is


$(u,v)(\overline{u},-v) = u\overline{u} - cv \overline{v}$.


It follows that if the norm form on $B$ is the Pfister form $<<a,b>>$, then the norm form on $C$ is the Pfister form $<<a,b,c>>$.


The resulting octonion aglebra $C$ depends not just $B$ but also on $c$. For example, suppose $B$ is the quaternion division algebra over $\mathbf{R}$ (i.e., Hamilton's quaternions). If one chooses $c=1$, then the resulting octonion algebra $C$ contains nonzero elements with norm $0$, while if one chooses $c = -1$ then the norm form on $C$ is anisotropic.


