There is a problem with this approach (I refer to Wadim Zudilin's answer). At least I don't see how to get from the first displayed equation (involving the $P_i$) to the second (involving the $L_i$) in his post.
Here is an example:
$(t^2+t-2)^2 + (3t+2)^2 + 1^2 = (t^2+t+2)^2 + (t+2)^2 + 1^2$
$(t_2+t_1-2t_0)^2 + (3t_1+2t_0)^2 + t_0^2 \ne (t_2+t_1+2t_0)^2 + (t_1+2t_0)^2 + t_0^2$,
by considering the coefficient of $t_0t_2$ or of $t_1^2$. Clearly, if the second equation were true it would imply the first. But the problem with the converse is that the first involves 5 degrees of freedom (the coefficients of $t^i$ with $0 \le i \le 4$), whereas the second has 6 (the coefficients of the $t_it_j$ with $i \le j$).
Edit: Here is a better example:
$(t^3+t)^2 (t^2+1)^2 -2 (t^2)^2 + t^2 t)^2 + 1^2 = (t^3-t)^2 t^2-1)^2 + 2 (t^2)^2 + t^2 t)^2 + 1^2$,
but the two forms $diag(1,-2,1,1)$ diag(1,-2,1)$ and $diag(1,2,1,1)$ diag(1,2,1)$ don't even have the same signature.