$B\langle i,j\rangle$ (awaiting review)

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 $a,b \in F$ with $i^2 = a$, $j^2 = b$, and $ji = -ij$. In this case, we write \[ B = B\langle i,j\rangle = \left(\frac{a,b}{F}\right). \]