Collecting together the earlier discussion on quaternions --- chronologically, I suppose this should be section 1.5. 7. The Mad Hatter's Tea Party _____________________________ After the reals \R ~= Cl(0,0), and complex numbers \C ~= Cl(0,1) generated by \I of dimension 2, come the quaternions \H ~= Cl(0,2) of dimension 4, equipped with their traditional basis \i,\j,\k, where \i \i = \j \j = \k \k = -1, \i \j = -\j \i = \k, \j \k = -\k \j = \i, \k \i = -\i \k = \j; [aka the Mad Hatter, the March Hare, the Dormouse; but killing Time, who thereupon took scalar offence.] A potential source of confusion arises from the resemblance between these definitions and those for Clifford generators \u,\v,\w (say) of Cl(0,3). There is a fundamental distinction between the concepts, the basis {1,\i,\j,\k} being linearly independent, but algebraically dependent: therefore only two members --- say \i,\j --- are available as generators of Cl(0,2), \k = \i\j mapping to a grade-2 versor. As a consequence, the threefold symmetry of the basis is lost under homomorphism to this graded algebra. A more satisfactory isomorphism of quaternions for our purposes embeds it as the even subalgebra within Cl(3,0) [or Cl(0,3) would do instead]: \H ~= Vl(0,3)^0 ~= Vl(3,0)^0 = Cl(3,0)^0, defined by \i -> \z\y, \j -> \x\z, \k -> \y\x. This map both preserves symmetry, and maps quaternions to versors; they are furthermore bi-vectors (grade-2), which proves to be the natural representation for rotations. In contrast are the complex bi-quaternions \H (x) \C of dimension 8, quaternions with components in \C --- or vice-versa, or indeed 2x2 complex matrices). These are isomorphic to the entire Clifford algebra \H (x) \C ~= \C (x) \H ~= \C(2) ~= Cl(3,0) under \i -> \z\y, \j -> \x\z, \k -> \y\x, \I -> \x\y\z. Not being restricted to versors, the embedding is only of restricted computational utility; on the other hand, the versor subgroup Vl(3,0) extends quaternions by one coset of (odd) reflections to the full rotation group of the sphere, the point group for Euclidean 3-space. WFL