Orientation preserving Möbius transformations in R∞4 and quaternionic determinants

This paper explores M(R∞4), the group of orientation preserving Möbius transformations acting in R∞4. On the one hand M(R∞4) is given by the group of 2×22×2 matrices over the quaternions HH with determinant DD derived from the corresponding 4×44×4 matrices over the complex numbers CC. On the other hand we know that, in general, M(R∞n) may be given in terms of 2×22×2 matrices over the Clifford algebra CnCn with n−1n−1 generators. Thus when n=4n=4, M(R∞4) is given in terms of 2×22×2 matrices with entries drawn from C4C4 and determinant Δ defined in terms of the entries of the given matrix. We note that the skew field HH may be considered as a Clifford algebra C3C3 based on two generators i and j   or more generally i1i1 and i2i2 where ij=kij=k or i1i2=ki1i2=k, while the set of elements {1,i,j,k}{1,i,j,k} form a basis of HH regarded as a 4-dimensional real vector space. Thus HH is embedded in C4C4. In the paper we reconcile the two representations of M(R∞4) by comparing the generating sets of the underlying groups of matrices. A relationship between a determinants DD and Δ is also exposed.