*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?