Root graded groups revisited

A group G is called root graded if it has a family of subgroups \( G_\alpha \) $G_{\alpha }$ indexed by roots from a root system \(\Phi \) $\mathrm{\Phi }$ satisfying natural conditions similar to Chevalley groups over commutative unital rings. For any such group there is a corresponding algebraic structure (commutative unital ring, associative unital ring, etc.) encoding the commutator relations between \( G_\alpha \) $G_{\alpha }$ . We give a complete description of varieties of such structures for irreducible root systems of rank \( \geqslant 3 \) $⩾3$ excluding \( {\textsf {H}}_3 \) ${\mathsf{H}}_3$ and \( {\textsf {H}}_4 \) ${\mathsf{H}}_4$ . Moreover, we provide a construction of root graded groups for all algebraic structures from these varieties.