  We study the purely leptonic decays of $W^\pm \to e^\pm e^\pm \mu^\mp \nu$
and $\mu^\pm \mu^\pm e^\mp \nu$ produced at the LHC, induced by sterile
neutrinos with mass $m_N$ below $M_W$ in the intermediate state. Since the
final state neutrino escapes detection, one cannot tell whether this process
violates lepton number, what would indicate a Majorana character for the
intermediate sterile neutrino. Our study shows that when the sterile neutrino
mixings with electrons and muons are different enough, one can still
discriminate between the Dirac and Majorana character of this intermediate
neutrino by simply counting and comparing the above decay rates. After
performing collider simulations and statistical analysis, we find that at the
$14~\text{TeV}$ LHC with an integrated luminosity of $3000~\text{fb}^{-1}$, for
two benchmark scenarios $m_N$ = 20 GeV and 50 GeV, at least a $3\sigma$ level
of exclusion on the Dirac case can be achieved for disparities as mild as e.g.
$|U_{Ne}|^2 < 0.7~ |U_{N\mu}|^2$ or $|U_{N\mu}|^2 < 0.7~ |U_{N e}|^2$, provided
that $|U_{Ne}|^2$, $|U_{N\mu}|^2$ are both above $\sim 2\times 10^{-6}$.

High Energy Physics - Phenomenology