Let $\mathfrak{sl}_2$ be the Lie algebra of traceless order two matrices with the usual basis $\{e,h,f\},$ such that $[e,f]=h, [h,e]=2e$ and $[h,f]=-2f$. I would like to expand the expression $(e+f)^n$, for $n\in\mathbb N$ in the universal enveloping algebra $U(\mathfrak{sl}_2)$.
I tried to consider the formulas coming from Baker–Campbell–Hausdorff, like the famous Zassenhauss formula. The Zassenhaus formula is $$e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],Y]) } \cdots$$ from this Wikipedia page. It is easy to see that some terms in this expression with $X=e$ and $Y=f$ are zero, for instance $[[[X,Y],X],X]$, $[[[X,Y],Y],Y]$ or any more general expressions containing sequences of three X or Y, like $[[[...X,],X],X]$, are always zero. However, I have not found a way write down an expression for $(e+f)^n$.
Is there any known formula for this? Anyone has some tips?