  The $\lambda$ mechanism ($W_L$-$W_R$ exchange) of the neutrinoless double
beta decay ($0\nu\beta\beta$-decay), which has origin in left-right symmetric
model with right-handed gauge boson at TeV scale, is investigated. The
revisited formalism of the $0\nu\beta\beta$-decay, which includes higher order
terms of nucleon current, is exploited. The corresponding nuclear matrix
elements are calculated within quasiparticle random phase approximation with
partial restoration of the isospin symmetry for nuclei of experimental
interest. A possibility to distinguish between the conventional light neutrino
mass ($W_L$-$W_L$ exchange) and $\lambda$ mechanisms by observation of the
$0\nu\beta\beta$-decay in several nuclei is discussed. A qualitative comparison
of effective lepton number violating couplings associated with these two
mechanisms is performed. By making viable assumption about the seesaw type
mixing of light and heavy neutrinos with the value of Dirac mass $m_D$ within
the range $1~ \textrm{MeV} < m_D < 1~ \textrm{GeV}$, it is concluded that there
is a dominance of the conventional light neutrino mass mechanism in the decay
rate.
