Let $D$ be a composition algebra over a field $F$. Define on $C = D \oplus D$ a product by,
$$(x,y)(u,v)=(xu + \lambda \bar{v}y, vx+y \bar{u}) $$
and a quadratic form $N:C \to F$ by
$$N((x,y))= N(x) - \lambda N(y)$$ where $0 \neq \lambda \in F.$
Prove that if $D$ is associative then $C$ is a composition algebra and $C$ is associative iff $D$ is commutative and associative.
For the first part I think I need to show that $N$ is multiplicative and the bilinear form $<(x,y),(u,v)> := N((x,y)+(u,v)) -N((x,y))-N((u,v))$ is nondegenerate but I'm not sure how to do so. For the second part, I hoped that (assuming associativity of $C$ and) setting $[(x,y)(u,v)](z,w) = (x,y)[(u,v)(z,w)]$, expanding, and then setting the 1st and 2nd coordinates equal to each other would force $D$ to be commutative and associative but this doesn't seem to be working.