  The lattice Boussinesq (lBSQ) equation is a member of the lattice
Gel'fand-Dikii (lGD) hierarchy, introduced in \cite{NijPapCapQui1992}, which is
an infinite family of integrable systems of partial difference equations
labelled by an integer $N$, where $N=2$ represents the lattice Korteweg-de
Vries (KdV) system, and $N=3$ the Boussinesq system. In \cite{Hiet2011} it was
shown that, written as three-component system, the lBSQ system allows for extra
parameters which essentially amounts to building the lattice KdV inside the
lBSQ. In this paper we show that, on the level of the Lagrangian structure,
this boils down to a linear combination of Lagrangians from the members of the
lGD hierarchy as was established in \cite{LobbNijGD2010}. The corresponding
Lagrangian multiform structure is shown to exhibit a `double zero' structure.
