  We prove (under certain assumptions) the irreducibility of the limit
$\sigma_2$ of a sequence of irreducible essentially self-dual Galois
representations $\sigma_k: G_{\mathbf{Q}} \to
\mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic
sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus \rho \oplus
\chi$ with $\rho$ irreducible, two-dimensional of determinant $\chi$, where
$\chi$ is the mod $p$ cyclotomic character. More precisely, we assume that
$\sigma_k$ are crystalline (with a particular choice of weights) and
Siegel-ordinary at $p$. Such representations arise in the study of $p$-adic
families of Siegel modular forms and properties of their limits as $k\to 2$
appear to be important in the context of the Paramodular Conjecture. The result
is deduced from the finiteness of two Selmer groups whose order is controlled
by $p$-adic $L$-values of an elliptic modular form (giving rise to $\rho$)
which we assume are non-zero.
