Formula for universal quadratic functor of direct product

For direct product of two groups
Suppose $$G$$ and $$H$$ are (possibly isomorphic, possibly non-isomorphic) abelian groups. Denote by $$\Gamma$$ the universal quadratic functor and denote by $$G \oplus H = G \times H$$ the external direct sum, which is the same as the external direct product, of $$G$$ and $$H$$. (Note that for abelian groups, direct products are also written as direct sums). Then, we have:

$$\Gamma(G \oplus H) = \Gamma(G) \oplus \Gamma(H) \oplus (G \otimes H)$$

where $$G \otimes H$$ denotes the tensor product of abelian groups.

For direct product of finitely many groups
Suppose $$G_1,G_2,\dots,G_n$$ are abelian group. Denote by $$\Gamma$$ the universal quadratic functor and denote by $$\bigoplus_{i=1}^n G_i$$ the external direct sum, which is the same as the external direct product, of all the groups. Then, we have:

$$\Gamma(\bigoplus_{i=1}^n G_i) = \left(\bigoplus_{i=1}^n \Gamma(G_i) \right) \oplus \bigoplus_{1 \le i < j \le n} (G_i \otimes G_j)$$