Let $B$ and $C$ be abelian groups (in additive notation). We call a function $f:B\rightarrow C$ a quadratic form if for all $x,y,z \in B$, the function $f$ satisfies the relation $f(x+y+z)-f(x+y)-f(y+z)-f(x+z)+f(x)+f(y)+f(z)=0.$
Is there some reference which says that it satisfies the parallelogram law $f(x+y)+f(x-y)=2f(x)+2f(y)$ and that for $B=\mathbb{Z}^n$ we can write $f(\displaystyle\sum_{i=1}^{n} a_i e_i)=\displaystyle\sum_{i=1}^n\big(2a_i^2+\displaystyle\sum_{j=1}^n a_ia_j \big) f(e_i)+ \displaystyle\sum_{1\leq i
Thank you!