Cubic Jordan algebras are not a series

Abstract

The idea of the exceptional series is that the exceptional simple Lie algebras should form a series. Since all four simple Lie algebras in the fourth row of the Freudenthal magic square are exceptional it is natural to ask if the remaining rows form a series. A stronger version of this question is that, for the first two rows (corresponding to the real and complex numbers), there is a category defined by a presentation which is a reasonable candidate for the series. Our main results show that neither of these candidates is a series but each consists of a finite set of points. In each case the series is defined by a parameter and we show that the relations imply that this parameter satisfies a polynomial. These two results were obtained by a computer calculation. Our calculation is supported by a website for inspection, and the calculations are certified by Lean 4.