The relative second Fox and third dimension subgroup of arbitrary groups

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Let Iᵣ(G) denote the augmentation ideal of the group algebra R(G) of a group G with coefficients in a commutative ring R. We give a complete description of the third relative dimension subgroup G∩(1+Iᵣ(K)Iᵣ(G)+I³ᵣ(G)) and the second relative Fox subgroup G∩(1+Iᵣ(K)Iᵣ(H)+I²ᵣ(G)Iᵣ(H)) for any subgroups K and H of G.