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We consider $p$-adic families of Siegel eigenforms of genus $2$ and finiteslope, defined as local pieces of an eigenvariety and equipped with a suitableintegral structure. Under some assumptions on the residual image, we show thatthe image of the Galois representation associated with a family is big, in thesense that a Lie algebra attached to it contains a congruence subalgebra ofnon-zero level. We call Galois level of the family the largest such level. Weshow that it is trivial when the residual representation has full image. Whenthe residual representation is a symmetric cube, the zero locus defined by theGalois level of the family admits an automorphic description: it is the locusof points that arise from overconvergent eigenforms for $\mathrm{GL}_2$, via a$p$-adic Langlands lift attached to the symmetric cube representation. Ourproof goes via the comparison of the Galois level with a "fortuitous"congruence ideal, that describes the zero- and one-dimensional subvarieties ofsymmetric cube type appearing in the family. We show that some of the $p$-adiclifts are interpolated by a morphism of rigid analytic spaces from aneigencurve for $\mathrm{GL}_2$ to an eigenvariety for $\mathrm{GSp}_4$. Theremaining lifts appear as isolated points on the eigenvariety.

Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms