Orbits of the maximal complex subgroup in the real torus

Question

Let $\Gamma$ be a discrete subgroup of $\mathbb C^n$ such that $Span_\mathbb C\ \Gamma=\mathbb C^n$ (i.e $\Gamma$ is a lattice of a maximal rank). Let $V:=Span_\mathbb R \ \Gamma$ and $ K:=V/\Gamma$ be the real torus.

Let $W:=V\cap iV$.How to show that $W$-orbits are dense in $K$?