If $X,Y$ are (local) vector fields on a Riemannian manifold $(M,g)$. Is there any bound, formula, estimate,... for the length of the Lie bracket, i.e., $g([X,Y],[X,Y])$?
For example, consider the $n$-dimensional sphere $\mathbb{S}^n$, and a local orthonormal frame $\{e_i\}$, i.e., $g(e_i,e_j)=\delta_j^i $. What can we say about $\|[e_i,e_j]\|^2$?
Is it possible to find an orthonormal frame such that $\|[e_i,e_j]\|^2=1$ for every $i,j$?
In general, such a bound cannot exist. This can be seen by "turning around" the problem: As stated in the comment by @Andrew_Hwang, the Lie bracket is independent of the metric. On $\mathbb R^n$ for $n\geq 3$ you can easily find local vector fields $X$ and $Y$, such that $X(x)$, $Y(x)$ and $[X,Y](x)$ are linearly independent for each $x$. This means that you can prescribe a Riemannian metric in such ah way that $X$ and $Y$ have constant norm $1$, while the norm of $[X,Y]$ is given by any positive smooth function. This can then be transferred locally to any manifold of dimension $\geq 3$.
The question whether you can arrange frames in such way that you get bounds on the Lie bracket seems to be of different nature. In a way, the Lie bracket then measures how far your orthonormal frame is from a coordinate frame, so this is measured by curvature. This should allow one to obtain a reasonable condition.