Cayley-Dickson Construction

The Cayley-Dickson construction is a generalization of the construction of the Complex Numbers to an arbitrary Algebra with Involution ( A , ⋅ , ( − ) † ) 𝐴 ⋅ superscript † (A,\cdot,(-)^{\dagger}) over a Involutive Ring R 𝑅 R .

We start by fixing some parameter γ : R : 𝛾 𝑅 \gamma:R ; we then proceed to define a multiplication operation on A ⊕ A direct-sum 𝐴 𝐴 A\oplus A like so:

(a,b)⁢(c,d)=(a⁢c−γ⋅d†⁢b,d⁢a+b⁢c†)𝑎𝑏𝑐𝑑𝑎𝑐⋅𝛾superscript𝑑†𝑏𝑑𝑎𝑏superscript𝑐†(a,b)(c,d)=(ac-\gamma\cdot d^{\dagger}b,da+bc^{\dagger})

We can also define an involution on A ⊕ A direct-sum 𝐴 𝐴 A\oplus A :

(a,b)†=(a†,−b)superscript𝑎𝑏†superscript𝑎†𝑏(a,b)^{\dagger}=(a^{\dagger},-b)