Abstract Let 𝔽 = ℝ, ℂ or ℍ. A p-set of equi-isoclinic n-planes with parameter λ in 𝔽r is a set of p n-planes spanning 𝔽r each pair of which has the same non-zero angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$. It is known that via a complex matrix representation, a pair of isoclinic n-planes in ℍr with angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2n-planes in ℂ2r with angle arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$. In this article we characterize all the p-tuples of equi-isoclinic planes in ℂ2r which come via our complex representation from p-tuples of equiangular lines in ℍr. We then construct all the p-tuples of equi-isoclinic planes in ℂ4 and derive all the p-tuples of equiangular lines in ℍ2. Among other things it turns out that the quadruples of equiangular lines in ℍ2 are all regular, i.e. their symmetry groups are isomorphic to the symmetric group S4.

中文翻译：

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四元等角线

摘要 让𝔽 = ℝ、ℂ 或ℍ。参数为 λ 在 𝔽r 中的等斜 n 平面的 p 集是一组 p n 平面跨越 𝔽r，每对具有相同的非零角反余弦 λ $\begin{array}{} \sqrt{ \lambda} \end{array}$。已知通过复矩阵表示，在 ℍr 中的一对等斜 n 平面与角 arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$ 产生一对等斜 2n- ℂ2r 中的平面，角为 arccos λ $\begin{array}{} \sqrt{\lambda} \end{array}$。在本文中，我们表征了 ℂ2r 中等斜面的所有 p 元组，这些 p 元组来自于我们对 ℍr 中等角线的 p 元组的复表示。然后我们在ℂ4中构造等斜面的所有p-元组，并在ℍ2中导出所有等角线的p-元组。