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started quadratic form
I understand that a quadratic form is a function $q\colon V \to k$ such that $q(t v) = t^2 q(v)$, and $v, w \mapsto q(v + w) - q(v) - q(w)$ is bilinear. This agrees with your definition if $2$ is invertible, but in general one cannot reconstruct $\langle{,}\rangle$.
Right, thanks. I said something to this extent, but it was a mess and it was wrong. I have fixed it now.
I added word polarization.
added historical pointer at the beginning of the References
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