The Algebra of the Paradoxical Topological Unity (S³ ↪ B⁴ )

The algebraic part of the PTU continues the series of Framework papers [10.5281/zenodo.23340020; 10.5281/zenodo.23438646]. The groups of manifolds are consistent with the embedding S³ ↪ B⁴ on the constructive ground (the Hopf fibration sequence) and the restrictive ground (the Adams theorem). The triple isomorphism S³≅ SU(2)≅Sp(1) is consistent with the quaternionic ball: B4 ≅ { q ∈ ℍ:∣q∣ ≤ RB } given ℝ⁴ ≅ ℍ through the connecting homomorphism ∂ and the locking of scale in SU(2). This missing operator is the quaternionic dilation r. The nontrivial part of the PTU is given by the multiplication table of three directions. A fundamental asymmetry of the left and right families of transformations has been discovered in the Framework: exp(t·i ·1/2) ≄ exp(t·i ). The algebra in B⁴: the fiber of the field is ℍ; the structure group of the fiber is SO(4) with the splitting so(4) ≅ su(2) ⊕ su(2). The lower discrete stage of the fibration sequence (Z₂)