Clifford Spinors and Root System Induction: $$H_4$$ and the Grand Antiprism

Recent work has shown that every 3D root system allows the construction of a correponding 4D via an `induction theorem'. In this paper, we look at icosahedral case $H_3\rightarrow H_4$ in detail and perform calculations explicitly. Clifford algebra is used to group theoretic based on versor theorem Cartan-Dieudonn\'e theorem, giving simple Pin Spin covers. Using connection with $H_3$ induction sheds light geometric aspects $H_4$ (the $600$-cell) as well other related polytopes their symmetries, such famous Grand Antiprism snub 24-cell. The uniform systems from procedure splitting respect subrootsystems into separate invariant sets further systematic insight underlying geometry. All are performed even subalgebra Cl(3), including Coxeter plane, which for visualising complementary pairs polytopes, shared supplementary computational sheets. This approach therefore constitutes more general way performing concerning groups, particular reflection groups systems, algebraic framework.