Let $A,X\in M_n(\mathbb{R})$. We denote $[A,X]=AX-XA$ the commutator. It is indeed a Lie Bracket for the matrix Lie algebra.
Taking $A$ constant, I'm looking for "the flow" of the commutator. That is to say, a "natural" function $\Phi([A,\cdot{}],t)$ with $t\in\mathbb{R}$ such that if $t=n\in\mathbb{N}$:
$$\Phi([A,\cdot{}],n)(X) = [A,\;\dots\;[A, [A,X]]\dots\;]=[(A)^n,X]$$
If such a function exists, is there a "good" way to "extend it" for a matrix $A$ changing continuously as the composition goes on ? That is, for $A:t\mapsto A(t)\in \mathcal{C}([t_0,t_f],M_n(\mathbb{R}))$ being a continuous function.
To sum up my question:
For $A$ constant: $$\Phi([A,\cdot{}],t)(X) = [(A)^t,X]=\;?$$
And for a continuous matrix $A$: $$\left(\mathop{\bigcirc}\limits_{t_0}^{t}\Phi([A(s),\cdot{}],ds)\right)(X)=\left(\mathop{\bigcirc}\limits_{t_0}^{t}[(A(s))^{ds},\cdot]\right)(X)=\;?$$
Thank you in advance for your kind help.
EDIT: Here is my main idea for the constant case so far:
First let's distinguish the exponential of a function and the exponential of a matrix:
$$\begin{aligned}\exp(f)&=Id+f+\frac{1}{2!}f\circ f+\frac{1}{3!}f\circ f\circ f+...\\e^{M}&=I_n+M+\frac{1}{2!}M^2+\frac{1}{3!}M^3+...\end{aligned}$$
It is known that $\exp([A,\cdot{}])=e^{A}\times\cdot{}\times e^{-A}$, by which I mean that for all $X\in M_n(\mathbb{R})$, we have $\exp([A,\cdot{}])(X)=e^{A}Xe^{-A}$. Hence:
$$\Phi(\exp([A,\cdot{}]),t)=\Phi(e^{A}\times\cdot{}\times e^{-A},t)=e^{tA}\times\cdot{}\times e^{-tA}$$
Now, one can imagine that:
$$\exp(\Phi([A,\cdot{}],t))=\Phi(\exp([A,\cdot{}]),t)$$
If one manages to show the above property and to extend the $\log$ operation to functions (when does it converge ?), we should obtain at the end:
$$\begin{aligned}\Phi([A,\cdot{}],t)&= \log(\exp(\Phi([A,\cdot{}],t))) \\ &=\log(\Phi(\exp([A,\cdot{}]),t))\\ &=\log(\Phi(e^{A}\times \cdot{}\times e^{-A},t))\\ &=\log (e^{tA}\times \cdot{} \times e^{-tA}) \\ &=\;? \end{aligned}$$
BUT, this approach might be very wrong, since we have: $$\Phi(\exp([A,\cdot{}]),t)=e^{tA}\times\cdot{}\times e^{-tA}=\exp([tA,\cdot{}])$$ and unfortunately I think we are faced with: $$\exp([tA,\cdot{}]) \neq\exp(\Phi([A,\cdot{}],t))$$
I would also like to extend the previous result to a more general setting where $f$ denotes a function taken without a lot of hypothesis:
$$\Phi(\exp{f},t)=\exp(t\times f)$$
Any reference or homemade proof on this matter would be very much appreciated: I will be happy to grant the bounty just for this result. I believe it can be useful for the non-constant case, if we get there at some point...