Abstract
The reductive pair ( B 3, G 2) over an arbitrary field F of characteristic≠2,3 is described in terms of an octonion algebra O over F and its associated spin representation. The reductive algebra associated with ( B 3, G 2) is shown to be isomorphic to the vector Malcev algebra of O . This is applied to realize the sphere S 7 as the reductive homogeneous space Spin(7)/ G 2 in an algebraic framework, and then to determine all invariant affine connections on S 7=Spin(7)/ G 2 in terms of the compact Malcev algebra of dimension 7. An application is also noted in reference to Lagrangian mechanics on S 7.
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