  We introduce and study the notion of "surface decomposable" variety, and
discuss the possibility that any projective hyper-K\"ahler manifold is surface
decomposable, which would produce new evidence for Beauville's weak splitting
conjecture. We show that surface decomposability relates to the
Beauville-Fujiki relation, a constraint on the cohomology ring of the variety,
and that general varieties with $h^{2,0}\not=0$ are not surface decomposable.
We also formalize the notion of triangle variety that is useful to produce
surface decomposition. We show the existence of these geometric structures on
most explicitly constructed classes of projective hyper-K\"ahler manifolds of
Picard number $1$.
