Split octonion algebra

Definition
A split octonion algebra over a field $$K$$ is an defining ingredient::octonion algebra (i.e., an 8-dimensional defining ingredient::composition algebra) $$A$$ over $$K$$ with norm $$N$$ such that there exists an element $$a \in A, a \ne 0$$ satisfying $$N(a) = 0$$.

For any field $$K$$, any two split octonion algebras are isomorphic as $$K$$-algebras (does the isomorphism also preserve the norm?). Hence, we can talk of the split octonion algebra over $$K$$.