A geometric characterisation of subvarieties of the standard E6-variety related to the ternions, degenerate split quaternions and sextonions over arbitrary fields
Abstract
The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety (the standard projective variety associated to the split exceptional group of Lie type E6) over an arbitrary field K. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions over K (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal-Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square.
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