But we can also swerve off this road and define the split octonions, \(\mathbb{O}'\),
to consist of pairs of quaternions with the modified product $$ (a,b)(c,d) = (ac + d\overline{b}, \; \overline{a}d + cb) $$

and conjugation defined as before. If we define $$ Q(x) = x \overline{x} $$ we still have $$ Q(xy) = Q(x) Q(y) $$ but now \(Q\) is a quadratic form of signature \((4,4)\), since $$ Q(a,b) = a \overline{a} - b \overline{b} $$ for any pair of quaternions \(a,b\).