I have a function $f(v_{1}, v_{2}): (\mathbb{R}^{3} \times \mathbb{R}^{3}) \rightarrow \mathbb{R}$ which maps two vectors in 3D space to a scalar. I know that this function is invariant under simultaneous rotations of both vectors about an arbitrary axis with arbitrary angle. More formally let $R \in {\rm SO}(3)$ (i.e. $R$ is a rotation matrix) then I know $f(Rv_{1}, Rv_{2}) = f(v_{1}, v_{2})$. Does it therefore follow that $f$ only depends on the rotational invariants in the system, i.e. $$f = f(|v_{1}|, |v_{2}|, \theta_{1,2}) = f(v_{1} \cdot v_{1}, v_{2} \cdot v_{2}, v_{1} \cdot v_{2})$$ where $v_{1} \cdot v_{2} = |v_{1}||v_{2}|\cos(\theta_{1,2})$, $|v|$ denotes the length of a vector and $\theta_{1,2}$ is the relative angle between $v_{1}$ and $v_{2}$.
Does this then generalise to a function of $N$ 3-vectors: $f(v_{1}, v_{2}, ..., v_{N})$. I.e. if $$f(Rv_{1}, Rv_{2}, ..., Rv_{N}) = f(v_{1}, v_{2}, ..., v_{N})$$ does it follow that $f$ is reducible to a function of all $\frac{1}{2} N (N+1)$ inner products between its arguments (which specify all rotational invariants in the system).