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) $.