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 $$\textrm{G2}$$ G2 from calibrations using $$\mathrm{Pin(7)}$$ Pin ( 7 ) . This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of $$\mathrm{Spin(}7)$$ Spin ( 7 ) that enables $$\textrm{G2}$$ G2 and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of $$\textrm{G2}$$ G2 in a way that breaks the symmetry of octonions. The 4-form calibration terms of $$\mathrm{Spin(7)}$$ Spin ( 7 ) are related to an ideal with three idempotents and provides a direct construction of $$\textrm{G2}$$ G2 for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of $$\textrm{G2}$$ 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.