  Let $G$ be an amenable group. We define and study an algebra
$\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal
subgroups of $G$. For a just infinite amenable group $G$, we show that
$\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group,
and in the case that it is nilpotent we determine the index of nilpotence. We
next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group
$G$, and show that it always contains nilpotent left ideals of arbitrarily
large index, as well as non-nilpotent elements. This provides infinitely many
finitely-generated counterexamples to a question of Dales and Lau, first
resolved by the author in a previous article, which asks whether we always have
$(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$. We further study this
question by showing that $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0
\}$ imposes certain structural constraints on the group $G$.
