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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.

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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.

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  • $\begingroup$ On the recent status of whether $S^6$ has a complex structure see Is there a complex structure on the 6-sphere? and Complex structure on $S^6$ gets published in Journ. Math. Phys on Math Overflow. Lots of controversy on whether or not it has been solved with some prominent mathematicians (like Atiyah) offering proofs that apparently turned out to be incorrect. $\endgroup$ – Conifold May 1 '17 at 18:23
  • $\begingroup$ @Conifold It is comforting to know that even mathematicians of Atiyah's quality can get it wrong.The most fiendishly difficult questions should result in the juiciest controversies - possibly because so few people are in a position to fully understand the material. $\endgroup$ – Nick R May 1 '17 at 18:42
  • $\begingroup$ Thanks for answering Nick, and sorry for the late response. Could you add links to the relevant sources to make this answer nice and complete? $\endgroup$ – Danu May 18 '17 at 18:50
  • $\begingroup$ @Danu Unfortunately, Atiyah's "potted" history doesn't provide specifics. I'll have a look around and see if I can locate the relevant sources. $\endgroup$ – Nick R May 18 '17 at 21:01
  • $\begingroup$ @Danu I've edited my answer to include some links to the original sources. I hope this is of some help. $\endgroup$ – Nick R May 18 '17 at 22:26

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