$[\nu_4,\iota_4]=?$, where $\nu_4$ is the Hopf map in $\pi_7(S^4)$ and $\iota_4$ is the generator of $\pi_4(S^4)$ which represents the identity map, $[,]$ is the Whitehead product.

This is how far I can get without checking a book. Since $v_4 = \pm {1\over 2} [i_4, i_4]$, you are interested in the triple Whitehead product $\alpha = {1\over 2}[ [i_4, i_4],i_4]$. Since it is a Whitehead product, its suspension is trivial. The Jacobi identity for Whitehead products shows that $[ [i_4, i_4],i_4]$ has order $3$; so $\alpha$ has order either $3$ or $6$. 

