Quaternion algebra (awaiting review)

Let $F$ be a field. A quaternion algebra over $F$ is a central simple algebra of dimension 4 over $F$.

A quaternion algebra $B$ over a field $F$ of characteristic not equal to 2 has elements $i,j \in B$ such that $\{1,i,j,ij\}$ is an $F$-basis for $B$, and such that there exists $a,b \in F^\times$ with $i^2 = a$, $j^2 = b$, and $ji = -ij$. In this case, we write \[ B = \left(\frac{a,b}{F}\right). \] We set $k = ij$.

We say that a quaternion algebra $B$ over $F$ is split if $B$ is isomorphic to the matrix algebra $\operatorname{M}_2(F)$, and we say $B$ is nonsplit otherwise.

If $B$ is a quaternion algebra over $\Q$, we say that $B$ is indefinite if $B\otimes_\Q \R$ is a split quaternion algebra over $\R$, and we say $B$ is definite otherwise.