Let $h:\pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ be the Hopf invariant. I believe that in the same paper that proves his suspension theorem, Freudenthal proved that if $x \in \pi_{2n-1}(S^n)$ satisfies $h(x)=0$, then $x$ is in the image of the suspension map $\pi_{2n-2}(S^{n-1}) \rightarrow \pi_{2n-1}(S^n)$. Observe that the usual Freudenthal suspension theorem says that the map $\pi_{2n-1}(S^n) \rightarrow \pi_{2n}(S^{n+1})$ is a surjection.

Can anyone either describe a proof of this or point me in the direction of a modern source for it? The only one I know of is Pontryagin's book "Smooth manifolds and their applications in homotopy theory", where this is Theorem 16, but I find his proof very hard to follow. I guess I would particularly appreciate a proof in the spirit of Pontryagin's proof, though more algebraic proofs are welcome too.