Split-quaternions

Remarks

The split-quaternions are a quaternion algebra over $\R$ that is not a division algebra, unlike Hamilton's quaternions. For example $(1 + j)(1 - j) = 1 - j^2 = 0$. They are isomorphic to the ring of $2 \times 2$ real matrices, by $i \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ and $j \mapsto \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Under this map the norm $a^2 + b^2 - c^2 - d^2$ becomes the determinant. So they form a composition algebra whose norm has signature $(2, 2)$. They are the Cayley–Dickson double of $\C$ with parameter $\mu = 1$.