This question derived from after I reading the book Naive Lie Theory, by Jhon Stillwell .
For the CBH thoerem, I am wondering how to construct some example about Exponential of Matrix, which has some property like : $$F_1(X,Y)=X+Y, F_2=\frac{1}{2}[X,Y],F_k[X,Y]=0,\forall k\geq 3$$ where $F_i(X,Y)$ comes from : $$e^Xe^Y=e^Z,Z=F_1(X,Y)+F_2(X,Y)+...$$ And here, the answer 1, give a example for scalar matrix, but I wonder if we can construct a function matrix (suppose a function matrix respect to variable $t$) also satisfying my require.
In deed, my question comes from the existence and uniqueness of solution of linear ODEs.
Any hints and help we be appriciated, thanks.
According to BCH, if $[X,Y]$ commute with $X$ and $Y$, then $\exp(X)\exp(Y)=\exp(X+Y+1/2[X,Y])$.