Let $A$ be a symmetric $d\times d$ matrix with integer entries such that the quadratic form $Q(x)=\langle Ax,x\rangle, x\in \mathbb{R}^d$, is non-negative definite. For which $d$ does it imply that $Q$ is a sum of finitely many squares of linear forms with integer coefficients $$ Q(x)=\sum_{i=1}^N (\ell_i(x))^2\quad \text{for some}\, N? $$ For $d=2$ this is true, I know it from the problem proposed by Sweden to IMO in 1995, but probably all this stuff is known for a longer time.

I think, I may prove it for some other small dimensions, although not so elementary (using Minkowski theorem on lattice points in convex bodies: if $Q$ is positive definite, we may find a linear form $\ell(x)$ such that $Q-\ell^2$ is still non-negative definite, this is equivalent to finding an integer point in an ellipsoid), but for large $d$ this argument fails.