On lie powers of regular modules in characteristic 2

We study Lie powers of regular modules for finite groups over a field of characteristic 2. First we prove two rather general reduction theorems, and then we apply them to Lie powers of the regular module for the Klein four group. For the latter, we solve the decomposition problem for the Lie power in degree 8, a module of dimension 8160. It has been known that of the infinitely many possible indecomposables, only four occur as direct summands in Lie powers of degree not divisible by 4, but that a fifth makes its appearance in the Lie power of degree 4. It is quite a surprise that no new indecomposables appear among the direct summands in degree 8.