Let $A$ be an algebra over $k$, $\operatorname{tr\_A}(x, y):=\operatorname{tr}(m\_{xy})$ be a trace form on $A$, and $V\_A$ be its restriction on the orthogonal complement to $1$. I wonder why a map $A \mapsto V\_A$ gives a bijection
$$\left\{ \begin{array}{cc} \text{central simple algebras over } k \\ \text{ of dimension } 4 \end{array} \right\} \leftrightarrow \left\{ \begin{array}{cc} \text{quadratic forms of rank } 3 \\ \text{ with discriminant } 4^3 \end{array} \right\}.$$


Any help or reference is welcome!



Reference request: correspondence between central simple algebras and quadratic forms
Everything is in Lam's book *Introduction to Quadratic Forms over Fields*. Theorem III 5.1 says: 


All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\right)$. 


Note that as quadratic space, $A = \langle 1, -a, -b, ab\rangle$ and $V\_A = \langle -a, -b, ab\rangle$.


Then Theorem III 2.5 says: 


Two quaternion algebras $A, A'$ are isomorphic as $k$-algebra iff $V\_A, V\_{A'}$ are isomorphic as quadratic spaces. 


On the other hand, all quadratic forms of rank three with $d=1$ ($=4^3$ in $k^\*/(k^\*)^2)$) are of the form $\langle c, d, cd\rangle$. Thus the one-to-one correspondence. 


