The radical of the bidual of a Beurling algebra

We prove that the bidual of a Beurling algebra on $\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that $\mathrm{r}\mathrm{a}\mathrm{d}({\mathrm{\ell }}^1({\oplus }_{i=1}^{\mathrm{\infty }}\mathbb{Z})\mathrm{"})$ contains nilpotent elements of every index. Each of these results settles a question of Dales and Lau. Finally we show that there exists a weight $\omega$ on $\mathbb{Z}$ such that the bidual of ${\mathrm{\ell }}^1(\mathbb{Z},\omega )$ contains a radical element which is not nilpotent.