Projective planes over 2-dimensional quadratic algebras

The split version of the Freudenthal–Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [5], [20] and [21]. The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A2×A2A2×A2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of “projective planes over 2-dimensional quadratic algebras”, in cases of the split and non-split Galois extensions, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.