The Gysin sequence for $M_{2,-1}$

Assume the Axiom of Choice. For $(h,j)=(2,-1)$ the Euler class of ${\xi }_{2,-1}$ is $e=(2-1)u=u$ by the class formula (Euler and first Pontryagin classes of ${\xi }_{h,j}$). The integral Gysin sequence of the oriented $S^3$-bundle $S^3\to M_{2,-1}\to S^4$ is $\cdots \to H^{k-4}(S^4;\mathbb{Z})\stackrel{⌣u}{\to }{H}^k(S^4;\mathbb{Z})\to H^k(M_{2,-1};\mathbb{Z})\to H^{k-3}(S^4;\mathbb{Z})\to \cdots ,$ and outside degrees $0$ and $4$ the base cohomology vanishes. The only nonempty multiplication map is $H^0(S^4)=\mathbb{Z}\stackrel{⌣u}{\to }{H}^4(S^4)=\mathbb{Z}$, multiplication by $1$, which is an isomorphism; exactness therefore forces $H^0(M_{2,-1};\mathbb{Z})=H^7(M_{2,-1};\mathbb{Z})=\mathbb{Z}$ and $H^k(M_{2,-1};\mathbb{Z})=0$ for $1\le k\le 6$. Transferring to homology by the universal coefficient theorem and finite generation gives $H_k(M_{2,-1};\mathbb{Z})=H_k(S^7;\mathbb{Z})$ for every $k$, which is exactly the statement verified by the homology-seven-sphere theorem (Euler number $\pm 1$ implies the Milnor sphere bundle is a homology seven-sphere).