Given a finite dimensional real vector space $V$ with two inner products $\langle,\rangle_g$ and $\langle,\rangle_{\tilde{g}}$, what are the inequalities for angles between vectors? Is there a similar inequality as for norms like \begin{align} a\lVert v\rVert_g\leq\lVert v\rVert_{\tilde{g}} \leq b\lVert v\rVert_g\,? \end{align} For instance, I'm looking for something like \begin{align} a \theta(v,w)\leq \tilde\theta(v,w)\leq b\theta(v,w)\,, \end{align} where $v,w\in V$ and $\theta(v,w)$ is the angle between the two vectors with respect to the inner product $g$ and $\tilde\theta(v,w)$ is the angle between the two vectors with respect to the inner product $\tilde g$.
This should be some standard knowledge, but I don't know a reference where this is shown/discussed. Does anybody have a tip for me?
Given two positive definite quadratic forms $g$ and $\tilde g$ on the real vector space ${\mathbb R}^2$ there is always a basis that diagonalizes both simultaneously. Therefore we may assume $$[g]=\left[\matrix{1&0\cr 0&1\cr}\right],\qquad [\tilde g]=\left[\matrix{\lambda&0\cr 0&\mu\cr}\right]$$ with $\lambda$, $\mu>0$. If it is only about angles you may even assume $\lambda\mu=1$, or something similar that suits your needs.
Using this setup it should not be too difficult to compute the maximal angle distortion in terms of $\lambda$ and $\mu$, and then universally. Unfortunately there is no universal upper bound.