Simple double Lie algebras on the Laurent polynomial space

We construct a simple $\lambda$-double Lie algebra on $k[t,t^{-1}]$ for every nonzero $\lambda \in k$. We then show that the analogous two-sided construction of weight zero is also simple. Both products admit natural coefficient realizations via row-and-column-finite operators associated with the finitary $\mathbb{Z}\times \mathbb{Z}$ matrix algebra.