$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

Question

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the conic which support is given by $x^2+y^2-z^2=0$. Let $G(\mathcal{C})$ be the subgroup of $\text{PGL}_2(\mathbb{K})$ composed by the automorphisms of the conic (they map the conic in itself). Prove that $\text{SO}(2,1)$ (the subgroup of $\text{GL}_3(\mathbb{K})$ that fix the quadratic form $q(x,y,z)=x^2+y^2-z^2$) is isomorphic to $G(\mathcal{C})$.