In [this paper of Mimura and Toda](http://projecteuclid.org/euclid.kjm/1250524818), tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi\_3(SU(4))$ as $i\_\ast \varepsilon\_3$. Here $i\colon SU(2) \hookrightarrow SU(4)$ is the inclusion such that $SU(4)/SU(2) \simeq S^5\times S^7$. Here $\varepsilon\_3$ should be some known element in $\pi\_3(SU(2)) = \pi\_3(S^3) = \mathbb{Z}$, but I can't figure out from information in the paper what it is (there are other elements labelled $\varepsilon$, but they live elsewhere with no obvious analogues in this dimension). Any ideas?


To contrast, the generator of $\pi\_3(SU(3))$ is given $i\_\ast\iota\_3$, where here $i\colon SU(2) \hookrightarrow SU(3)$ is the inclusion in the top left corner, so that $SU(3)/SU(2) \simeq S^5$, and $\iota\_3$ is the isomorphism $S^3\stackrel{\sim}{\to} SU(2)$.


I know that the generator of $\pi\_3$ of (suitable) Lie groups is given by an inclusion of $SU(2)$ as a subgroup, constructed involving longest roots in the Lie algebra, how do I reconcile this description with that in the first paragraph?



Generator of $\pi_3(SU(4))$ in Mimura-Toda
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.


