Eisenstein congruences for SO(4,3), SO(4,4), spinor and triple product L-values

We work out instances of a general conjecture on congruences between Hecke eigenvalues of induced and cuspidal automorphic representations of a reductive group, modulo divisors of certain critical L-values, in the case that the group is a split orthogonal group. We provide some numerical evidence in the case that the group is SO(4,3) and the L-function is the spinor L-function of a genus 2, vector-valued, Siegel cusp form. We also consider the case that the group is SO(4,4) and the L-function is a triple product L-function.