There is a very natural way to define generators of $\pi\_{4n-1}(SO(4n))\cong \mathbb{Z}\oplus \mathbb{Z}$ in terms of quaternions when $n=1$ and octonions when $n=2$ (see for example Tamura, *On Pontrjagin classes and homotopy types of manifolds*, 1957). Since there are no normed division algebras in higher dimensions, it is not possible to do the same for $n>2$.


I was wondering whether there still is a natural identification between $\pi\_{4n-1}(SO(4n))$ and $\mathbb{Z}\oplus\mathbb{Z}$ using explicit generators when $n>2$?



Homotopy groups $\pi_{4n-1}(SO(4n))$
One way to think about this would be to consider the (injective!) map $\pi\_{4n-1} SO(4n) = \pi\_{4n} BSO(4n) \to \mathbf Z^2$ that sends the classifying map of a $4n$-dimensional oriented real vector bundle $\xi\colon E \to S^{4n}$ to the pair $(e,p\_n)$ consisting of its Euler and $n$th Pontryagin class evaluted against the fundamental class of the base sphere.


This map is injective and for $n \neq 1,2,4$ its image is given by $2\mathbf{ Z} \times a\_n(4n-1)!\mathbf{Z}$ where $a\_n = 2$ if $n$ is odd, and $1$ otherwise: the Euler number of any bundle is even by the Hopf invariant $1$ problem, and the possible values of $p\_n$ of (stable) vector bundles over$ S^{4n}$ is known from homotopy theory, see Theorem 3.8. in these [notes](https://math.mit.edu/juvitop/old/notes/2014_Fall/JPLEVINElectures.pdf) by Levine, for instance.


The generators you are looking for are then simply given by two bundles that are sent to $(2,0)$ and $(0,a\_n(2n-1)!)$. The first one is clearly the tangent bundle of $S^{4n}$. I am not sure if the second one, which has vanishing Euler class and minimal $p\_n$, has a nice geometric description.


