TITLE: Computation of $\pi_4$ of simple Lie groups
QUESTION [13 upvotes]: Below we assume any simple Lie group $G$ to be simply connected.
$\pi_3(G)=\mathbb{Z}$ for any simple Lie group $G$ and there is a uniform proof for that.
Now the textbooks say $\pi_4(G)$ is trivial except for $G=Sp(n)$, for which it is $\mathbb{Z}/2\mathbb{Z}$. 
My question is the following: is there a uniform way to derive this fact, in such a way to show which property of the symplectic group is so special?

REPLY [3 votes]: There is actually a collection of more general statements that detail the structure of the $\pi_4$.
Firstly it was Browder [The Cohomology of Covering Spaces of H-Spaces] who proved that a simply connected finite H-space is 2-connected. Clark [On $\pi_3$ of Finite Dimensional H-Spaces] showed that the first non-vanishing homotopy group of a finite loop space appeared in degrees 1 or 3 (the result that $\pi_2=0$ for a simply connected finite H-space also appears in Browders [Torsion in H-Spaces], see also Lin's [The First Homotopy Group of a Finite H-Space]). Kane and Hubbuck [On $\pi_3$ of a Finite H-Space] prove that $\pi_3$ is torsion free, building on previous work of Thomas and Lin which required stricter assumptions (the Lie group $E_8$ does not fall under Thomas's requirements, for instance).
Finally Hubbuck [Homotopy Groups of Finite H-Spaces] studied $\pi_4$, showing that it was a direct sum of finite groups of order 2. The proof uses JHC Whitehead's Certain Exact Sequence (see Baues [Homotopy Type and Homology] for a good discussion), identifying the groups that appear within using the work of others. The end result is an exact sequence
$H_5(X;\mathbb{Z})\xrightarrow{\rho_2} H_5(X;\mathbb{Z}_2)\xrightarrow{Sq^2_*}H_3(X;\mathbb{Z}_2)\rightarrow \pi_4X\rightarrow0$
And we know that $H_3(X;\mathbb{Z}_2)=\oplus\,\mathbb{Z}_2$ from the Hurewicz theorem and the work of Kane and Hubbuck so the result on $\pi_4X$ follows.
Harper claims that if $X$ is simply connected and $H_*(\Omega X;\mathbb{Z})$ is torsion free then the result follows from the previous work of Bott and Samelson [Applications of the Theory of Morse to Symmetric Spaces].