Classification-free proof that all reflections arise from lines in the exceptional quaternionic line systems

Establish a proof, independent of Cohen’s classification and without computer assistance, that for the star-closed line system L associated with each of Cohen’s exceptional quaternionic reflection groups of rank at least three, every reflection in the corresponding unitary reflection group W equals r_ell for some line ell in L (i.e., is the reflection across the hyperplane orthogonal to a line in L).

Background

The paper studies line systems associated with quaternionic reflection groups and shows that, for the exceptional groups of rank at least three classified by Cohen, every reflection is realized as the order‑2 reflection r_ell across the hyperplane orthogonal to a line ell in the specified line system L. This fact is verified using Cohen’s classification together with computer checks.

A conceptual proof avoiding reliance on the classification and computational verification is not provided, and the authors explicitly note the lack of such a proof. Establishing a direct, classification‑free proof would clarify the structural relationship between reflections and the underlying line systems for these exceptional quaternionic groups.