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 $\mathbb{R}{\mathrm{P}}^{11}$.