A group G is called root graded if it has a family of subgroups Gα indexed by roots from a root system Φ 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 Gα. We give a complete description of varieties of such structures for irreducible root systems of rank ⩾3 excluding H3 and H4. Moreover, we provide a construction of root graded groups for all algebraic structures from these varieties.