Real spin bundles over $\mathbb{C}\mathrm{P}^3$ and a new Euclidean embedding of $\mathbb{R}\mathrm{P}^7$

We generalize the $\alpha$-invariant introduced by Atiyah and Rees to an invariant of real spin bundles and use it to classify real bundles over $\mathbb{C}\mathrm{P}^3$ admitting spin structure. We apply this result to show that $\mathbb{R}\mathrm{P}^7$ can be smoothly embedded in $\mathbb{R}\mathrm{P}^{11}$.