  We present a recently developed theory for the inclusive breakup of
three-fragment projectiles within a four-body spectator model
\cite{CarPLB2017}, for the treatment of the elastic and inclusive non-elastic
break up reactions involving weakly bound three-cluster nuclei in $A\,(a,b)\,X$
/ $a = x_1 + x_2 + b$ collisions. The four-body theory is an extension of the
three-body approaches developed in the 80's by Ichimura, Autern and Vincent
(IAV) \cite{IAV1985}, Udagawa and Tamura (UT) \cite{UT1981} and Hussein and
McVoy (HM) \cite{HM1985}. We expect that experimentalists shall be encouraged
to search for more information about the $x_{1} + x_{2}$ system in the elastic
breakup cross section and that also further developments and extensions of the
surrogate method will be pursued, based on the inclusive non-elastic breakup
part of the $b$ spectrum.
