Kac moody root spaces are finite dimensional

Quotient each homogeneous component of F2 by its intersection with $\mathfrak{r}$. The split graded ideal has no zero component, so the resulting zero part is $\mathfrak{h}$ and all remaining parts have strictly positive or strictly negative degree. Every element still has finite support.

Every bracket of length $m$ is a linear combination of right-nested brackets of length $m$: apply $[[u,v],w]=[u,[v,w]]-[v,[u,w]]$ repeatedly to reduce the left bracket length. There are $n^m$ choices of letters for such a bracket. Thus the total height-$m$ subspace of either half, and hence each of its degree quotients, has dimension at most $n^m$.

Independence of the roots makes a lattice point on $\mathbb{C}{\alpha }_i$ an integer multiple of ${\alpha }_i$. A bracket word at positive degree $k{\alpha }_i$ uses only $e_i$; all words of length $k>1$ vanish since $[e_i,e_i]=0$. Degree ${\alpha }_i$ is spanned by $e_i$ and is nonzero since $[e_i,f_i]=h_i\mathrm{\ne }0$. The negative statement follows in the same way. If $\beta >0$ is not ${\alpha }_i$ and its simple reflection is a root, some coefficient at $j\mathrm{\ne }i$ is positive and remains unchanged by the reflection; the one-sign property forces the reflected root to stay positive.