Assembling comments of Neil Strickland and Allen Knutson, we have that $\varepsilon_3$ is just the standard generator $\iota_3\colon S^3 \stackrel{\sim}{\to} SU(2)$, since the inclusion $SU(2) \to SU(4)$ induces an isomorphism on $\pi_3$. This is shown using the long exact sequence in homotopy groups, and the fact $S^5\times S^7$ is 3-connected.