that the open subset of $N$ given by the non-vanishing of the 3 frozen variables $x_4, x_5, x_6$ is equal to $$N^{w_0} := N \cap (B_-w_0B_-),$$ where $B_-$ denotes the subgroup of lower triangular matrices in $G$, and $w_0$ is the longest element of the Weyl group of $G$. This is an example of a *unipotent cell*, that is, a stratum of the decomposition of the unipotent group $N$ obtained by intersecting it with the Bruhat decomposition of $G$ associated with the opposite Borel subgroup $B_-$. The coordinate ring $\C[N^{w_0}]$ is obtained by localizing the polynomial ring $\C[N]$ at the element $x_4x_5x_6$, and we can see that it carries almost the same cluster algebra structure as $\C[N]$, the only difference being that the *coefficient ring* generated by the frozen cluster variables is now the Laurent polynomial ring in $x_4, x_5, x_6$.

Lie theory
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We now introduce a class of varieties generalizing the varieties $N$ and $N^{w_0}$ of Example \[sect2.2\]. These are subvarieties of unipotent subgroups of symmetric Kac-Moody groups, and we first prepare the necessary notation and definitions. For more details, we refer to [@Ku].

Kac-Moody Lie algebras {#sect3.1}
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Let $C = [c_{ij}]$ be a symmetric $n\times n$ 