By Jacques Dixmier

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Making the obvious identification of vectors f1, w with imaginary quaternions v, w, the real part of the quaternionic product vw is then (minus) the dot product f1. W, while the imaginary part is the cross product f1 x W. 3) is clearly a vector space isomorphism, it induces an isomorphism relating the linear maps on these spaces. 3) in 2-component quaternionic language. 7) since the left-hand side is the 4 x 4 identity matrix and the right-hand side is the 2 x 2 identity matrix. 2). 7), as expected.

The volume element of the (n + 1) -dimensional paravector space is the product of the paravector basis elements, in which every second element is Clifford-conjugated. It is, thus, equal to the volume element eT of Gin to within a sign. 4) of a general element in Gin. 6 Slices of biparavectors and paravectors dual to triparavectors The slice of a simple biparavector B by the subspace normal to the paravector direction ell is the paravector (Bell)Re since, as shown above, this lies in B and is orthogonal to ell .

Clifford Algebras and their Applications in Mathematical Physics © Springer Science+Business Media New York 2000 22 Tevian Dray and Corinne A. Manogue on an equal footing. We then consider in Section 4 the massless Dirac equation on MajoranaWeyl spinors (in momentum space) in 10 dimensions, which can be nicely described in terms of 2-component spinors over the octonions 0, the only other normed division algebra besides R, C, and 1HI. Solutions of this equation are automatically quaternionic and thus lend themselves to the preceding quaternionic description.