# How to bound the expectation value of a commutator?

Are there formal ways to bound the following quantity:

$$\langle[[{S_x},{H}],{H}]\rangle$$

The expectation value is taken on an eigenstate of $$S_x$$.

$$H$$ is a dipolar Hamiltonian acting on $$N$$ distinguishable spin-3 particles. $$S_x=\sum_i^NS_x^i$$ is the sum of $$N$$ spin-3 operators acting on distinguishable particles. $$H$$ and $$S_x$$ do not commute.

I am looking for formal and general ways to bound these quantities, so I do not think the details of the operators or the system are needed, let me know if I am wrong.