Notations and definitions: $\mathfrak g $= finite-dimensional complex simple lie algebra of rank $n$; $\mathfrak h$ = a fixed Cartan subalgebra; $R, R^+$ its set of roots and positive roots; $\alpha_1, \cdots, \alpha_n$ its simple roots and $\theta$ its highest root; $x_\alpha^\pm \in \mathfrak g$ and $h_\alpha$ such that the subalgebra generated by $x_\alpha^\pm, h_\alpha$ is ismorphic to $\mathfrak sl_2$; $L^e(\mathfrak g) = \mathfrak g\otimes \mathbb C[t,t^{-1}]\oplus \mathbb Cd$ the extended loop algebra, where $[d,x\otimes t^m] = m x\otimes t^m;$ $\mathfrak h^e = \mathfrak h \oplus \mathbb Cd$ the extended cartan subalgebra; $\delta\in (\mathfrak h^e)^*$ such that $\delta (\mathfrak h )= 0$ and $\delta(d)=1;$ by $x_m$ we denote the element $x\otimes t^m \in L^e(\mathfrak g)$ and set $e_i^\pm = x_{\alpha_i}^\pm\otimes 1, e_0^\pm = x_\theta^{\mp}\otimes t^{\pm1} $.
A representation $V$ of $U(L^e(\mathfrak g))$ is called integrable if $V = \bigoplus V_\lambda$, where $V_\lambda = \{v\in V: hv = \lambda(h)v \ \ \forall\ h\in \mathfrak h^e\}$ and $e_i^\pm$ act locally nilpotently on $V (\ \mbox{given } v\in V, (e_i^\pm)^nv = 0$ for some $n$).
Remark: All the notations and definitions above (and others) can be found in the very first pages of the cited paper below with many more details.
Question: For an integrable representation $V$, in this paper https://arxiv.org/abs/math/0004174, on page 4, it is stated that "It is well known that the elements $x_{\beta,m}^\pm$ act locally nilpotently on $V$ for all $\beta\in R^+, m \in \mathbb Z$." I want to verify this result by myself as I couldn't find it anywhere.
What I tought so far: Considering only $x_{\beta,m}^+$, the first thing I've noticed is that $x_{\beta,m}^+ V_\lambda \subseteq V_{\lambda + \beta +m\delta}$. Indeed, for $h\in \mathfrak h^e$, write $h = h_0 + cd$. Then, $[h, x_{\beta,m}^+] = [h_0+cd,x_{\beta}^+\otimes t^m] = \beta(h_0)x_\beta^+\otimes t^m + cm\ x_\beta^+\otimes t^m = [\beta(h)+m\delta(h)]x_\beta^+\otimes t^m$, where for $\beta \in \mathfrak h^*$ we can extended it to $(\mathfrak h^e)^*$ by setting $\beta(d)=0$. Thus, for any $0 \neq v\in V_\lambda$ we obtain $h(x_{\beta,m}^+v)= [\lambda(h)+\beta(h)+m\delta(h)]x_{\beta,m}^+.$
Therefore, we obtain $(x_{\beta,m}^+)^kV_\lambda \subseteq V_{\lambda + k(\beta + m\delta)}.$ At the same time, we can write $\beta = a_1 \alpha_1 +\cdots +a_n\alpha_n, a_i \in \mathbb N$. There are $\ell_1,\cdots,\ell_n \in \mathbb N$ such that $(e_i^+)^{\ell_i}V_\lambda \neq \{0\}$ and $(e_i^+)^{\ell_i+1}V_\lambda = \{0\}$ by the fact that $V$ is integrable. Somehow I want to relate these integers $\ell_i$ with the integer $k$ such that $(x_{\beta,m}^+)^kV_\lambda = \{0\}$, but I couldn't be succesful.
Any hint on how to prove this?