Construction of Exceptional Lie Algebra G2 and Non-associative Algebras Using Clifford Algebra

Abstract This article uses Clifford algebra of positive definite signature to derive octonions and the Lie exceptional algebra G2 G2 from calibrations using Pin (7) Pin (7). This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of Spin (7) Spin (7) that enables G2 G2 and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of G2 G2 in a way that breaks the symmetry of octonions. The 4-form calibration terms of Spin (7) Spin (7) are related to an ideal with three idempotents and provides a direct construction of G2 G2 for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of G2 G2 without using the Lie bracket. A calibration in 15 dimensions is shown to generate the sedenions and to include one of the power-associative algebras, a result previously found by Cawagas.