Editing Cayley–Dickson construction (section)

== Modified Cayley–Dickson construction ==
{{further|Split-complex number|Split-quaternion|Split-octonion|Bicomplex number|Biquaternion|Bioctonion}}
The Cayley–Dickson construction, starting from the real numbers <math>\mathbb R</math>, generates the [[composition algebra]]s <math>\mathbb C</math> (the [[complex number]]s), <math>\mathbb H</math> (the [[quaternion]]s), and <math>\mathbb O</math> (the [[octonion]]s). There are also composition algebras whose norm is an [[isotropic quadratic form]], which are obtained through a slight modification, by replacing the minus sign in the definition of the product of ordered pairs with a plus sign, as follows:
<math display="block">(a, b) (c, d) = (a c + d^* b, d a + b c^*).</math>

When this modified construction is applied to <math>\mathbb R</math>, one obtains the [[split-complex number]]s, which are [[ring isomorphism|ring-isomorphic]] to the [[direct product]] <math>\mathbb R \times \mathbb R;</math> following that, one obtains the [[split-quaternion]]s, an [[associative algebra]] [[isomorphic]] to that of the 2&thinsp;×&thinsp;2 real [[matrix (mathematics)|matrices]]; and the [[split-octonion]]s, which are isomorphic to {{math|[[Split-octonion#Zorn's vector-matrix algebra|Zorn('''R''')]]}}. Applying the original Cayley–Dickson construction to the split-complexes also results in the split-quaternions and then the split-octonions.<ref>Kevin McCrimmon (2004) ''A Taste of Jordan Algebras'', pp 64, Universitext, Springer {{ISBN|0-387-95447-3}} {{mr|id=2014924}}</ref>