The bivector Clifford algebra and the geometry of Hodge dual operators

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Abstract
This article describes features of a natural representation of the Clifford algebra of the space of bivectors of a four-dimensional vector space, the representation space being the vector space plus its dual. Vectors and covectors are then pure spinors. The natural map taking a pure spinor to a totally null 3-bivector is shown to be intertwined with the operation of lowering or raising by a metric with the action of the Hodge dual operator. New formulae exhibiting a metric in terms of its dial operator are presented, as are a few applications.

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