By now I'm probably talking to myself --- so just 5. Postscript _____________ The case a = 1, b = 0 of A surreptitiously introduced earlier can of course be cracked using the general factorisation given subsequently; however, if tackled in isolation there's no obvious algorithm available to unearth say A = 1 + \x\v - \y\w + \x\y\v\w = \x \y (\y - \w) (\x + \v) . This pathological horror is non-invertible, with magnitude zero; a = -1, b = 0 is the only other such case. It conjugates every vector remorselessly into zero (representing no object). More strangely still, it has complex conjugate rotation factors, even though the curve tangent formula would surely yield a (single) real factor for real S --- the explanation being that S = S^- is a double root, illustrating inter alia how this formula is inherently inapplicable to isoclinic cases! Of course, there's much more I could ramble on about --- proofs and algorithms, interactive Java movies of these isometries, plausible canards (I stopped counting at 12); the mapping from the algebra to the geometry; Bott periodicity; Dupin cyclides; a neat connection with linked spheres ... And then there are parabolic and isoclinic isometries, leading to an exhaustive classification of the conjugacy types of isometry. The last topic seems curiously neglected --- part of the reason may simply be that nineteenth century sources gave the impression that it had all been sorted out, at least in low dimension. Close inspection reveals that --- often without explicit mention --- they work over \C rather than \R, severely restricting any application to concrete, intuitive geometry; and no means of verifying correctness or completeness of any claimed classification is likely to be apparent. A more currently active area which has many obvious parallels with, and ought surely to bear some relevance to, these investigations is Lie algebra. Alas --- despite having embarked upon the consumption of numerous tracts upon this subject --- I have yet to acquire substantive understanding of what it might actually be useful for, and in particular of how it might be applied to the classification problem. Can anybody out there enlighten me? WFL