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?



Computation of $\pi_4$ of simple Lie groups
As a partial answer, here's at least a uniform statement, which can be found as Theorem 3.10 in Mimura's survey "Homotopy theory of Lie groups" in the Handbook of Algebraic Topology: 



> 
> Let $G$ be a compact, connected, simply connected, simple Lie group, $T$ a maximal torus of $G$, $\mathbb{R}^r$ its universal covering and $\Gamma$ the inverse image of the identity in $\mathbb{R}^r$. Let $a$ be the dominant root with respect to some lexicographic order of the roots of $G$. Then $\pi\_4(G)$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ if the hyperplane $a=1$ contains a point of $\Gamma$ and $0$ otherwise. 
> 
> 
> 


(In the survey article, the statement has a $\mathbb{Z}$ where I wrote $\mathbb{Z}/2\mathbb{Z}$ but that would appear to be a typo.)


Mimura refers to the paper


* R. Bott and H. Samelson. Applications of the theory of Morse to symmetric spaces. Amer. J. Math. 80 (1958), 964-1029.


I can't check right now, but the uniform statement suggests a uniform proof, relating the fourth homotopy group to information on the root system. It would seem that the difference that singles out the symplectic groups is that the long roots are divisible (by 2 incidentally) in the character group.


