Integer positive definite quadratic form as a sum of squares

This is a well-known problem, called the Waring's problem of integral quadratic forms. Every semi-positive definite quadratic form in $n \leq 5$ variables is a sum of $n + 3$ squares of linear forms. This was proved by Chao Ko, but this can be explained by the fact that the quadratic form of sum of $n + 3$ squares has class number 1 if $n \leq 5$.

There are positive definite quadratic forms in $n \geq 6$ variables which cannot be written as sums of squares of integral linear forms. The smallest example is the quadratic form corresponding to the root system $E_6$.

So, one should look at the set of positive definite quadratic forms in $n$ variables that can be written as sums of squares of linear forms. Then there exists an integer $g(n)$ such that all these quadratic forms can be written as a sum of $g(n)$ squares of integral linear forms. The magnitude of $g(n)$ is not known. The best upper bound is $O(e^{k\sqrt{n}})$ for some explicit $k$. This is obtained recently by Beli-Chan-Icaza-Liu (appeared in TAMS).