The Explicit Twisted Group Algebra Structure of the Cayley–Dickson Algebra

The Cayley–Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley–Dickson algebra as a twisted group algebra with an explicit twist function \(\sigma (A,B)\). We show that this function satisfies the equation

$$\begin{aligned} e_Ae_B=(-1)^{\sigma (A,B)}e_{A\oplus B} \end{aligned}$$

and provide a formula for the relationship between the Cayley–Dickson algebra and split Cayley–Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley–Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley–Dickson algebra and split Cayley–Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.