THE REDUCTIVE PAIR (B3, G(2)) AND AFFINE CONNECTIONS ON S7

Abstract

The reductive pair (B3, G2) over an arbitrary field F of characteristic not-equal 2,3 is described in terms of an octonion algebra O over F and its associated spin representation. The reductive algebra associated with (B3, G2) is shown to be isomorphic to the vector Malcev algebra of O. This is applied to realize the sphere S7 as the reductive homogeneous space Spin(7)/G2 in an algebraic framework, and then to determine all invariant affine connections on S7 = Spin(7)/G2 in terms of the compact Malcev algebra of dimension 7. An application is also noted in reference to Lagrangian mechanics on S7.