The space of orientation-compatible almost complex structures on the six-dimensional sphere naturally contains a copy of seven-dimensional real projective space. We show that the inclusion induces an isomorphism on fundamental groups and rational homotopy groups. We also compute the homotopy fiber of the inclusion and the homotopy groups of the space of almost complex structures in terms of the homotopy groups of the seven-dimensional sphere. Our approach lends itself to generalization to components of almost complex structures with vanishing first Chern class on six-manifolds.

We thank Luis Fernandez and Scott Wilson for useful comments, and the referee for helping improve the exposition; the third author likewise thanks Claude LeBrun and Dennis Sullivan. The second author was partially supported by FCT/Portugal through project UIDB/MAT/04459/2020.