We study zero divisors whose components alternate strongly pairwise and construct oriented hexagons in the zero divisor graph of an arbitrary real Cayley–Dickson algebra. In case of the algebras of the main sequence, the zero divisor graph coincides with the orthogonality graph, and any hexagon can be extended to a double hexagon. We determine the multiplication table of the vertices of a double hexagon. Then we find a sufficient condition for three elements to generate an alternative subalgebra of an arbitrary Cayley–Dickson algebra. Finally, we consider those zero divisors whose components are both standard basis elements up to sign. We classify them and determine necessary and sufficient conditions under which two such elements are orthogonal.

中文翻译：

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真实 Cayley-Dickson 代数的正交图。第一部分：双重替代零除数及其六边形

我们研究零除数，其分量强烈成对交替，并在任意实数 Cayley-Dickson 代数的零除数图中构造定向六边形。在主序列代数的情况下，零除数图与正交图重合，任何六边形都可以扩展为双六边形。我们确定双六边形顶点的乘法表。然后我们找到三个元素生成任意 Cayley-Dickson 代数的替代子代数的充分条件。最后，我们考虑那些成分都是标准基元素的零除数。我们对它们进行分类并确定两个这样的元素正交的必要和充分条件。