Let $F$ be a field. Consider quadratic forms $f=\sum_{1\leq i\leq n}x_i^2$ with $n=1,2,4,8$. $f$ is a group form if $\{d\in F^\star|\exists x\in F^n, f(x)=d\}$ is a group.
For $F=Q$, it follows from classical theorem that $\{d\in F^\star|\exists x\in F^n, f(x)=d\}$ does form a subgroup of $Q^+$.
$\textbf{Q:}$ In Lam's Introduction to Quadratic Forms Chpt 1, Sec 2, it says for $n=1,2,4,8$, $f$ is a group form for any field. From Wiki, it seems that it should hold for $n=16$ as well. Note that existence is easy by 2 squares. Then the multiplicative closureness follows from application of correponding $n$ square identities. Does one have any even sum of squares identities as https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity ?
$\textbf{Q':}$ Was there inductive argument of $n$ for such kind of identities?
If that is the case, then I deduce $f$ of the form above is a group form for any even number $n$ and $n=1$.