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It is known that the sphere $ S^6$ admits an almost complex structure by identifying $S^6 $ with the space of unit purely imaginary Cayley numbers.

I would like to show that this almost complex structure is not integrable using the Nijenhuis tensor.

Can someone explain to me how vector fields look like in $S^6$ and how can we apply them in the Nijenhuis tensor?

I found something in Ballmann's book (Kahler manifolds) but I didn't understand it.

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    Could you explain more details which part you do not follow in Ballmann's book? – 2017-02-26
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    How he defines the vector fields and how he calculates the tensor? Actually I need to understand everything in order to show it is not integrable – 2017-02-26

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