I am reading Neisendorfer's paper Samelson products and exponents of homotopy groups and related papers. I am stuck on theorem 14.1 on page 21, which says that there exists a $\mathbb{Z}_{p^{r+1}}$ summand in $\pi_{2p^kn - 1}(P^{2n+1}(p^r))$ for any $k \geq 1$ where $p > 3$ is a prime number and $P^{2n+1}(p^r) = D^{2n+1} \cup_{p^r} S^{2n}$.

The overall idea of the proof is to show that there exists an element in position $2p^kn - 2$ in the mod $p$ homotopy Bockstein spectral sequence for $\Omega P^{2n+1}(p^r)$ such that survives-until and then dies-in the $(r + 1)^\text{th}$ page. The first step is to find such an element and to show that it *survives*.

The method consists in comparing the mod $p$ homotopy Bockstein ss $E_\pi^*$ with the mod $p$ homology Bockstein ss $E_H^*$ via the mod $p$ Hurewicz map $\phi$. Both the spectral sequences are differential graded Lie algebras via, respectively, the Samelson product and the commutator of the Pontryagin product; moreover the Hurewicz map is a morphism of differential graded Lie algebras.

The mod $p$ homology Bockstein ss is simple. To see this, first notice that $H^*(P^{2n}(p^r); \mathbb{Z}) \cong \mathbb{Z}_{p^r}\langle \alpha \rangle$ where $\deg \alpha = 2n$, hence $H_*(P^{2n}(p^r); \mathbb{Z}) \cong \mathbb{Z}_{p^r}\langle\beta\rangle$ where $\deg \beta = 2n - 1$ and finally $H_*(P^{2n}(p^r); \mathbb{Z}_p) \cong \mathbb{Z}_p\langle u, v \rangle$ where $\deg u = 2n - 1$ and $\deg v = 2n$. Then we have $E^1_H \cong H_*(\Omega P^{2n+1}(p^r); \mathbb{Z}_p) \cong H_*(\Omega\Sigma P^{2n}(p^r); \mathbb{Z}_p) \cong \mathbb{Z}_p^\text{Alg} \langle u, v\rangle$ by the Bott-Samelson theorem. Moreover $\beta^s_H = 0$ for $s < r$ and $\beta^r_Hv = u$, so $E^1_H = \cdots = E^r_H$ and $E^{r+1}_H = \cdots = E^\infty_H \cong \mathbb{Z}_p$.

The mod $p$ homotopy Bockstein ss is not that simple, but surely we have at least two elements $\mu, \nu$ respectively in degrees $2n - 1$ and $2n$ such that $\phi\mu = u$ and $\phi\nu = v$ for the mod $p$ Hurewicz theorem. Moreover $\beta^s_\pi = 0$ for $s < r$ and $\beta^r_\pi v = u$, so $E^1_\pi = \cdots = E^r_\pi$.

Now the Lie algebra structures come to play. In any graded differential Lie algebra $(L, d)$ over a field of characteristic $p > 2$ any element $x \in L$ of degree $2n$ gives rise to some *cycles* $\sigma_k(x), \tau_k(x)$ in degrees $2p^kn - 2$ and $2p^kn - 1$ for any $k \in \mathbb{N}$. Moreover, in the universal enveloping algebra of $L$ holds $dx^{p^k} = \tau_k(x)$.

Specializing to our Bockstein sss, this gives us the cycles $\sigma_k(\nu)$, $\tau_k(\nu)$ in $E_\pi^*$ and $\sigma_k(v), \tau_k(v)$ in $E_H^*$. Moreover, since they are constructed via the Lie algebra structures, we have that $\phi\sigma_k(\nu) = \sigma_k(v)$ and $\phi\tau_k(\nu) = \tau_k(v)$. The claim is that $\sigma_k(\nu)$ survives until $E^{r+1}_\pi$ for any $k$ and $\tau_k(\nu)$ survives until $E^{r+1}_\pi$ for any $k > 1$.

How can I see the survival of $\sigma_k(\nu)$? Probably I am missing something stupid, but at the moment I cannot see it. Thanks in advance to anyone.