Defines integral planes O² with norm sum quadratic form over crystallographic and non-crystallographic orders, recovers root systems including E8⊕E8 and H4⊕H4, proves no indecomposable rank-8 golden octonion order exists, and shows the algebraic Hopf map restricts to a fibration on the balanced unit
Baez,The octonions, Bull
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Integral Planes and Unit-Norm Polytopes
Defines integral planes O² with norm sum quadratic form over crystallographic and non-crystallographic orders, recovers root systems including E8⊕E8 and H4⊕H4, proves no indecomposable rank-8 golden octonion order exists, and shows the algebraic Hopf map restricts to a fibration on the balanced unit