Use Theorem 8.9 from Whitehead's "Elements of Homotopy Theory" on page 537. Take $k=2$, $\beta=\iota_4$ and $\alpha\in\pi_8S^4$. Use $[\iota_4,\iota_4]=\Sigma\nu'-2\nu_4$, $h_0(\nu_4\circ\eta_7)=h_0(\nu_4)\circ\eta_8=\iota_7\circ\eta_8=\eta_8$ and $h_0(\Sigma\nu'\circ\eta_7)=h_0(\Sigma(\nu\circ\eta_6))=0$. The triple product disappears and you are left with David Frank's answer.
Just a reminder that $\pi_8S^4=\mathbb{Z}_2\{\Sigma\nu'\circ\nu_7\}\oplus\mathbb{Z}_2\{\nu_4\circ\eta_7\}$. The Hopf invariants and whitehead products (and some formulas to compute them) are available in Oguchi's paper "Generators of 2-Primary Components of Homotopy Groups of Spheres, Unitary Groups and Symplectic Groups".