On Lie powers of the augmentation ideal in characteristic 2

Let G be a finite group, K a field and V a finite-dimensional KG-module. Write L(V) for the free Lie algebra on V. The action of G on V extends naturally to L(V), so it becomes a KG-module which is direct sum of finite-dimensional submodules. For a finite-dimensional KG-module V, let I(V) denote the set of isomorphism classes of indecomposable direct summands of V, and I(L(V))=⋃n⩾1I(Ln(V)). It is natural to ask whether I(L(V)) is finite. Of course, this is not a question unless there exist infinitely many isomorphism classes of indecomposable KG-modules (that is, K has positive characteristic p and the Sylow p-subgroups of G are non-cyclic and ).Let K be a field of characteristic 2 and G the Klein four group. We write D=KG, Δ for the augmentation ideal of KG, Δ⁎ for the dual of Δ and V1, V2, V3 for the three 2-dimensional induced modules from the three cyclic subgroups of G. In this article we concentrate on Lie powers Ln(Δ) of Δ, and describe the module decomposition of L16(Δ), a module of dimension 2690010. In particular, I(L16(Δ))={D,V1,V2,V3,Δ⁎}.