Quaternionic equiangular lines

Let 𝔽 = ℝ, ℂ or ℍ. A p-set of equi-isoclinic n-planes with parameter λ in 𝔽r is a set of pn-planes spanning 𝔽r each pair of which has the same non-zero angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. It is known that via a complex matrix representation, a pair of isoclinic n-planes in ℍr with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$ yields a pair of isoclinic 2n-planes in ℂ2r with angle arccos $\begin{array}{} \sqrt{\lambda} \end{array}$. 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.