I have to deal with an expression of the form, $$(Ae+Bf+Ch)^n$$
where $n \geq 0$ and $A,B,C \in \mathbb C$ where $e,f,h$ is the usual Chevalley basis of $\mathfrak{sl}_2$, i.e. $$[e,f] = h, \quad [h,e] = 2e, \quad [h,f] = -2f.$$
$()^n$ is the $n$-fold product, which I take to make sense in $U(\mathfrak{sl}_2)$.
Ideally, I would like some expansion of $(Ae+Bf+Ch)^n$ in terms of products of the individual $e,f,h$. The reason for this is for my purposes it is useful if I can leverage that $e,f,h$ are individually primitive elements (in the co-algebra sense). Of course the term $(Ae + Bf+Ch)^n$ is not primitive.
I don't know if seeking a neat formula is foolish. The term will be some sum over words which you can re-arrange into your preferred ordering, so there might not be something convenient.