  We examine the lattice generated by two pairs of supplementary vector
subspaces of a finite-dimensional vector space by intersection and sum, with
the aim of applying the results to the study of representations admitting two
pairs of supplementary invariant spaces, or one pair and a reflexive form. We
show that such a representation is a direct sum of three canonical
sub-representations which we characterize. We then focus on holonomy
representations with the same property.
