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

On the Lagrangian multiform structure of the extended lattice Boussinesq system