Why does the unitary evolution of an operator involve a commutator with the Hamiltonian?

The equation for the unitary evolution of an operator $hat{A}$ is given (as far as I understand) as $$frac{partialhat{A}}{partial t}=frac{i}{hbar}[hat{H},hat{A}].quadmbox(1)$$

It seems that $(1)$ is related to the Baker-Campbell-Haussdorf via a lemma which is written as follows: $$mbox{Ad}_{e^X}Y=e^{mbox{ad}_X}Y=e^XYe^{-X}=Y+[X,Y]+frac{1}{2!}[X,[X,Y]]+frac{1}{3!}[[X,[X,[X,Y]]]+…quad (2)$$

I notice that e.g. $$frac{partial}{partial lambda}big(e^{lambda hat{H}}hat{A}e^{-lambda hat{H}}big)bigg|_{lambda=0}=[hat{H},hat{A}].$$

This makes $(1)$ look like an infinitesimal change in $t$ on the left-hand side, and some sort of infinitesimal Lie group action of $hat{H}$ on the right.

Is this interpretation correct?

Any explanations, especially good texts to consult, would be greatly appreciated.

Thanks in advance!