Alternative approach based on embedding the automorphism group?

[lukeburns]

I am wondering if there is an alternative starting point based on simultaneously embedding both a vector space and its automorphism group in the algebra. The reason I imagine this might offer a path forward is because without contraction, elements of the algebra could not be generators of transformations within the algebra.

[lukeburns]

http://geocalc.clas.asu.edu/pdf/LGasSG.pdf