  In this paper, we investigate the closure of a large class of Teichm\"uller
discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus
L of translation surfaces of genus three. We describe a systematic way to prove
that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of
L. The strategy of the proof is an analysis of completely periodic directions
on such a surface and an iterated application of Ratner's theorem to unipotent
subgroups acting on an ``adequate'' splitting. This analysis applies for
example to all Teichmueller discs stabilized obtained by Thurston's
construction with a trace field of degree three which moreover ``obviously not
Veech''. We produce an infinite series of such examples and show moreover that
the favourable splitting situation does not arise everywhere on L, contrary to
the situation in genus two. We also study completely periodic directions on
translation surfaces in L. For instance, we prove that completely periodic
directions are dense on surfaces obtained by Thurston's construction.
