A geometric connection between the split first and second rows of the Freudenthal-Tits magic square (Q6060068)

In this paper, the authors prove a remarkable connection between the varieties of the frist and the second row of the split Freudenthal-Tits magic square, over an arbitrary field. The connection was conjectured by the reviewer, and the authors did a wonderful job proving it. The result (and my original conjecture) is roughly the following: If a variety \(\mathcal{V}\) of the second row is contained in a non-degenerate quadric \(Q\) (required to be of maximal Witt index in case of the first column), then the tangent hyperplanes at the maximal (convex) quadrics (or \textit{symps} in the language of parapolar spaces) of \(\mathcal{V}\) that are contained in a singular subspace of \(Q\) form the dual of the corresponding variety in the first row. In other words, calling white points the points of the variety, grey points the points of the secant variety that are not white, and other points black, these hyperplanes form the set of tangent hyperplanes through a given black point. But the authors take it a step further and extend this to degenerate quadrics: they show that, up to projective equivalence, only two degenerate quadrics contain the given variety, and the tangent hyperplanes at the symps contained in singular subspaces define unique grey and white points. The proofs are rather geometric, and beautiful. An important ingredient is the classification of geometric hyperplanes of the varieties of the second row of the Square, available for the third and the fourth cell, but not for the second cell. This difficulty for the second cell had to be overcome.