  We investigate mixing of neutrinos in the $\nu$MSM (neutrino Minimal Standard
Model), which is the MSM extended by three right-handed neutrinos. Especially,
we study elements of the mixing matrix $\Theta_{\alpha I}$ between three
left-handed neutrinos $\nu_\alpha$ ($\alpha = e,\mu,\tau$) and two sterile
neutrinos $N_I$ ($I=2,3$) which are responsible to the seesaw mechanism
generating the suppressed masses of active neutrinos as well as the generation
of the baryon asymmetry of the universe (BAU). It is shown that $\Theta_{eI}$
can be suppressed by many orders of magnitude compared with $\Theta_{\mu I}$
and $\Theta_{\tau I}$, when the Chooz angle $\theta_{13}$ is large in the
normal hierarchy of active neutrino masses. We then discuss the neutrinoless
double beta decay in this framework by taking into account the contributions
not only from active neutrinos but also from all the three sterile neutrinos.
It is shown that $N_2$ and $N_3$ give substantial, destructive contributions
when their masses are smaller than a few 100 MeV, and as a results $\Theta_{e
I}$ receive no stringent constraint from the current bounds on such decay.
Finally, we discuss the impacts of the obtained results on the direct searches
of $N_{2,3}$ in meson decays for the case when $N_{2,3}$ are lighter than pion
mass. We show that there exists the allowed region for $N_{2,3}$ with such
small masses in the normal hierarchy case even if the current bound on the
lifetimes of $N_{2,3}$ from the big bang nucleosynthesis is imposed. It is also
pointed out that the direct search by using $\pi^+ \to e^+ + N_{2,3}$ and $K^+
\to e^+ + N_{2,3}$ might miss such $N_{2,3}$ since the branching ratios can be
extremely small due to the cancellation in $\Theta_{eI}$, but the search by
$K^+ \to \mu^+ + N_{2,3}$ can cover the whole allowed region by improving the
measurement of the branching ratio by a factor of 5.
