Root graded groups of type $H_3$ and $H_4$

Using the well-known realisation of the root system $H_4$ as a folding of $E_8$, one can construct examples of $H_4$-graded groups from Chevalley groups of type $E_8$. Such Chevalley groups are defined over a commutative ring $R$, and the root groups of the resulting $H_4$-grading are coordinatised by $R\times R$. We show that every $H_4$-graded group arises as the folding of an $E_8$-graded group, or in other words, that it is coordinatised by $R\times R$ for some commutative ring $R$. We also prove similar assertions for $(D_6,H_3)$ in place of $(E_8,H_4)$.