The 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 $$e_Ae_B=(-1)^{\sigma(A,B)}e_{A\oplus B}$$ 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.