  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.
