The opposite simple centralizer in a Kac Moody half vanishes

Decompose $x$ into finitely many weights. For fixed $i$, the brackets of its distinct components with $f_i$ have distinct weights, so each bracket is zero. It suffices to treat a homogeneous $x$ of positive degree $\beta$. Let $J$ be the span of $x$ and all iterated $\mathrm{ad}{e}_i$ applied to $x$. Every such vector has degree $\beta +\gamma$ with $\gamma \in Q^{+}$, so $J\subset {\mathfrak{n}}^{+}$.

The space $J$ is stable under all $e_i$ by construction and under $\mathfrak{h}$ by its homogeneous spanning vectors. Induct on the number of positive adjoints to prove stability under $f_i$. The base is $[f_i,x]=0$. For $y$ already treated, $[f_i,[e_j,y]]=-{\delta }_{ij}[h_i,y]+[e_j,[f_i,y]]$ lies in $J$. Thus $J$ is an ideal, and $J\cap \mathfrak{h}=0$ by its positive degrees. F1 forces $J=0$, hence $x=0$. The sign-changing involution interchanges the two conclusions.