  We compute explicitly the action of the group of affine diffeomorphisms on
the relative homology of two remarkable origamis discovered respectively by
Forni (in genus 3) and Forni-Matheus (in genus 4). We show that, in both cases,
the action on the non trivial part of the homology is through finite groups. In
particular, the action on some 4-dimensional invariant subspace of the homology
leaves invariant a root system of $D_4$ type. This provides as a by-product a
new proof of (slightly stronger versions of) the results of Forni and Matheus:
the non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the
Teichmuller disks of these two origamis are equal to zero.
