= Quaternion algebra
{title2=$(a,b)_k=k\langle i,j\rangle/(i^2-a,j^2-b,ij+ji)$}
For a <field> of characteristic other than two and nonzero $a,b$, the <quaternion algebra> has basis $1,i,j,ij$ with the displayed relations. It is a four-dimensional <central simple algebra>. After adjoining a square root of $a$, diagonalizing $i$ and taking $j=\left(\begin{smallmatrix}0&b\\1&0\end{smallmatrix}\right)$ identifies it with $M_2$. Quaternion conjugation fixes scalars and negates $i,j,ij$, giving norm $x_0^2-a x_1^2-b x_2^2+ab x_3^2$. An element is invertible exactly when its norm is nonzero. Thus the <quaternion algebra> is either a <division algebra> or a split <matrix algebra>. The familiar <quaternion> algebra is $(-1,-1)_{\mathbb R}$.
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