The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$. If I need to calculate some bracket like $[x_\beta^{(n)}, x_\alpha^{(m)}]$, how should I proceed? Can I proceed doing the calculation to $[x_\beta^{n}, x_\alpha^{m}]$ and then to multiply by $\frac{1}{n!m!}$ and try to rewrite the result in a divided power notation?

ADDED: Let $g$ be a finite-dimensional simple Lie algebra, $\{\alpha_i,...,\alpha_n\}$ the set of positive roots. It is known that there exist an integral form for $U(g)$ and it is generated by $(x_{\alpha,r}^-)^{(k)}$ where $k>0$, $r\in Z$ and $\alpha$ is a positive root. How to calculate $[(x_{\alpha,r}^-)^{(k)}, (x_{\beta,s}^-)^{(m)}]$?