  A future lepton collider, such as the proposed CLIC or ILC, would allow to
study top quark properties with unprecedented precision. In this paper, we
present a method to reconstruct the $t \bar{t}$ decay in the dilepton channel
at future $e^+e^-$ colliders. We derive a simple, closed analytical expression
for the neutrino four-momenta as a function of the $W$ boson mass and develop a
maximization procedure to find the optimal solution for the reconstruction of
the full $t \bar{t}$ event. We show that our method is able to reconstruct
neutrino four-momenta with an error of less than $2 \, \%$ in $60 \, \%$ of the
times. Finally, we test the performance of this reconstruction method in the
calculation of the helicity fractions of the $W$ boson. A precise measurement
of these observables could be used to probe new physics effects in the $Wtb$
vertex. We find that, from a large $t \bar{t}$ sample, our reconstruction
method allows to calculate these observables with an accuracy better than $1 \,
\%$.
