A note on invariants of foliated 3-sphere bundles

In this note we prove that $H^{\ast }(\text{BSO}(4);\mathbb{Q})$ injects into the group cohomology of ${\text{Diff}}^{+}(S^3)$ with rational coefficients. The proof is based on an idea of Nariman who proved that the monomials in the Euler and Pontrjagin classes are nontrivial in $H^{\ast }({\text{BDiff}}_{+}^{\delta }(S^{2n-1});\mathbb{Q})$.