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Our main aim is to provide a uniform geometric characterization of theanalogues over arbitrary fields of the four complex Severi varieties, i.e.~thequadric Veronese varieties in 5-dimensional projective spaces, the Segrevarieties in 8-di\-men\-sional projective spaces, the line Grassmannians in14-dimensional projective spaces, and the exceptional varieties of type$\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can beregarded as a far-reaching generalization of Mazzocca and Melone's approach tofinite quadric Veronesean varieties. This approach takes projective propertiesof complex Severi varieties as smooth varieties as axioms.

On the varieties of the second row of the split Freudenthal-Tits Magic Square