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3-Sphere and Lie group theories (Fig. 1, 2) enable calculations of the dihedral angle with right sign and stereochemistry from NMR data[1, 2]: carbon, proton chemical shift ( 1 H, 13 C [ppm]), and vicinal constant coupling ( 3 J HH [Hz]), and manifold equations transform NMR data in angles. The dihedral angles of the mixture 1-α:1-β-L-lyxose are calculated with Lambert R value under 3-sphere approach rule from the ratio of α H1 -H 3 H 4 :β H1 -H 3 H 4 , or with manifold equation from carbon or proton chemical shift under conic section. Six sets angles on two units, U1 and its parallel unit S1, are build with manifold angles under equilateral triangle rule. The coordinates[3] on R 4 (X 0 = cosϕsinθ HnHn+1 , X 1 = sinϕsinξ*/2, X 2 = cosθ HnHn+1 sinϕ, X 3 = sinθ HnHn+1 sinξ/2), confirmed by algebraic equations (cis-, trans-ee: 3 J HH = ϕ 1/2 /2, trans-aa: 3 J HH = ϕ 1/2 ), gives greet relationships between the dihedral angle and the vicinal angle. Scoop of the paper: Our aim is to demonstrated once again the gift of NMR data for calculation dihedral angles with 3-sphere approach. In the absence of one of the NMR data the manifold equations are solutions for all cases.

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