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

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.

由于缺乏明确的乘法表，凯利-迪克森代数长期以来一直是一个挑战。尽管可以通过归纳构造来构造，但其显式结构迄今为止仍然难以捉摸。在本文中，我们通过将凯利-迪克森代数揭示为具有显式扭曲函数 \(\ sigma (A,B)\)的扭曲群代数，提出了解决这个长期存在的问题的解决方案。我们证明这个函数满足方程

给出了Cayley-Dickson代数与分裂Cayley-Dickson代数之间关系的公式，从而给出了分裂Cayley-Dickson代数扭曲函数的显式表达式。我们的方法不仅解决了凯莱-迪克森代数和分裂凯莱-迪克森代数缺乏显式结构的问题，而且还揭示了这个基本数学对象背后的代数结构。