Who first wrote down $S^6$'s standard almost complex structure? And who first proved that it is not integrable?

Question

It is well known that $S^6$ admits an almost complex structure, inherited from its manifestation as the space of unit imaginary octonions. This almost complex structure is also well-known not to be integrable, i.e. not derived from a genuine complex structure.

Both of these facts were known by the early 50's, as mentioned e.g. in the introduction to this paper by Calabi, which contains references (numbers 8 and 23 for its existence and numbers 7, 9 for non-integrability).

I suspect that the existence might have been known for some decades by that time, since it already appeared in a textbook (the above-mentioned reference 23) and is a relatively easy linear algebra-type result. Perhaps Cartan wrote it down already?

Regarding non-integrability, I suspect (for no particularly strong reasons) that it was only discovered around 1950 (the references Calabi gives are from 1951), but I'm not sure.

Answer 1

According to this arxiv paper by Atiyah, existence and construction dates from 1947; non-integrability from 1951.

Here is Atiyah's history:

Ehresmann 1947: Introduced the notion of almost complex structure and showed that the 6-sphere admits an almost complex structure, but explicitly points out that he does not know whether it has a complex structure.

Hopf 1947: Proved that $S^4$ and $S^8$ do not admit almost complex structures.

Kirchhoff 1947: Uses octonions to construct an explicit almost complex structure on $S^6$.

Eckmann-Frohlicher and Ehresmann-Liberman 1951: Independently prove that Kirchhoff’s almost complex structure on $S^6$ is not integrable to a complex structure.

Borel and Serre 1953: Prove that $S^{2n}$ admits an almost complex structure if and only if $n = 1$ or $3$.

Hirzebruch 1954 and Liberman 1955: Remarks that it is still not known whether $S^6$ has a complex structure.

---

EDIT

Regarding your request for sources, here are my “mixed results”.

Ehresmann’s 1947 introductory results can be found here : Ehresmann, C., Sur la theorie des espaces fibres, Colloque de Topologie Algebrique, C.N.R.S., Paris (1947), pp. 3.

Borel and Serre’s non-existence result for even dimensional spheres can be found here : A. Borel, J.P. Serre Groupes de Lie et puissances réduites de Steenrod Amer. J. Math., 75 (1953), pp. 409-448

For Hopf’s 1947 result, I cannot locate the original paper, however the source is identified in Robert Green and Shing-Tung Yau’s Differential Geometry as H. Hopf, Sur les champs d’element de surface dans les varietes a 4 dimensions, Topologie Algebrique, Paris 1947, Editions CNRS.

The original paper giving Kirchhoff's construction is not easy to locate. Possibly from A. KIRCHHOFF, C. R. Acad. Sci. Paris vol. 225 (1947).

I am not able to source the non-integrable to a complex structure result.