I am trying to calculate the Chern class of the tangent bundle of the sphere $S^6$. I am told that this is an interesting case, since $S^6$ is not a complex manifold, but it has an almost complex structure on it, induced by the octonions (and I would like to learn more abut complex structures and the octonions). The Chern class of $S^2$ can be found by using its complex structure, but that can't be extended to $S^6$.

Apparently, one way of solving this is viewing $S^6$ in $\mathbb R^7$, then $\mathbb R^7$ as the "imaginary" part of the octonions, and the octonions as $\mathbb R^8$. With such a decomposition we could write $T\mathbb R^8|_{S^6}=TS^6\oplus L$ for $L$ a line bundle (the "real" part of the octonions), but I'm not sure how that works out. So my question is how I would find the Chern class through this complex structure approach.