Let $V$ be a 2-dimensional real vector space and $\langle\cdot,\cdot\rangle$ a non-degenerate inner product on $V$. Consider a function $f:SO(3)\times V\to V$ such that:
(1) for all $A\in SO(3)$, the map $f(A,\cdot)=f_A:V\to V$ is a linear isometry.
(2) $f_I$ is the identity map on $V$.
(3) $f(A\cdot B,v)=f\big(A,f(B,v)\big)$ for all $A,B\in SO(3)$ and $v\in V$.
If the following statement is true: given $u,v\in V$ with $\langle u,u\rangle=\langle v,v\rangle$ then there exists $A\in SO(3)$ such that $f_A(v)=u$, how could we conclude that $\langle\cdot,\cdot \rangle$ is positive definite?