Two advanced examples from the theory of abelian varieties (for example, elliptic curves) where it is non-trivial to prove that you even get a quadratic form.

Let $A/k$ be an abelian variety defined over an algebraically closed field. Then the degree map $$ \deg : \operatorname{End}(A)\longrightarrow\mathbb Z $$ is a quadratic form. A reference for this is Mumford's Abelian Varieties (or my Arithmetic of Elliptic Curves for the dimension 1 case). To indicate why this is non-trivial, I will mention that if you know it for $k=\overline{\mathbb F}_p$, then you can use basic facts about the Frobenius map (roughly, that $a+b\phi$ is separable if $p\nmid a$) to prove the Weil estimate for $\#A(\mathbb F_{p^n})$.

The second example is when $k$ is the algebraic closure of $\mathbb Q$. Let $D$ be a symmetric divisor on $A$. Then the canonical height function $$ \hat h_D : A(k) \longrightarrow \mathbb R $$ of Neron and Tate defined by $$ \hat h_D(P) = \lim_{n\to\infty} 4^{-n}h_D(2^nP) $$ is a quadratic form. Further, if $D$ is ample, then $\hat h_D(P)=0$ if and only $P$ is a point of finite order. Canonical heights are of great importance in studying the arithmetic of abelian varieties. For example, they appear prominently in the statement of the Birch-Swinnerton-Dyer conjecture. They are discussed in Lang's Fundamentals of Diophantine Geometry and my book with Hindry Diophantine Geometry: An Introduction. (Again, the elliptic curve case is covered in my AEC.)