Split quaternions H_s

The real split quaternion algebra is $\mathbb H_s=M_2(\mathbb R)$ , with ordinary matrix multiplication, conjugation $\overline X=\operatorname{tr}(X)I_2-X$ , and multiplicative quadratic form $N(X)=\det X$ . It is a four-dimensional real associative algebra.

Quaternion basis

Take

i=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad j=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad k=ij.

Then $i^2=-1$ , $j^2=k^2=1$ , and $ij=-ji$ . In this basis, $N(a+bi+cj+dk)=a^2+b^2-c^2-d^2$ , of signature $(2,2)$ .

Distinction from Hamilton quaternions

The nonzero matrices $\operatorname{diag}(1,0)$ and $\operatorname{diag}(0,1)$ multiply to zero. Consequently this algebra is not the Hamilton division algebra. The split form and $M_2(\mathbb R)$ have separate construction records joined by a specified isomorphism.

References

Alberto Elduque, Composition algebras, Section 2.1, equation (7); Theorem 2.11 and Corollary 2.12.