A class of continuous non-associative algebras arising from algebraic groups including $E_8$

We give a construction that takes a simple linear algebraic group $G$ over a field and produces a commutative, unital, and simple non-associative algebra $A$ over that field. Two attractions of this construction are that (1) when $G$ has type $E_8$, the algebra $A$ is obtained by adjoining a unit to the 3875-dimensional representation and (2) it is effective, in that the product operation on $A$ can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.

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