For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex structures $I,J,K$ and so three decompositions $$ T = T^{(1,0)}_a \oplus T^{(0,1)}_a, ~~~~~~ \textrm{ for } a = I,J,K. $$ The three almost complex structures are required to give a representation of the quaternions: one asks that $$ IJK = -1. $$ What does this explicitly mean for the three decompositions of $T$? I guess there is a simple way to see this condition in terms of the intersections of the various subspaces . . .
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4$\begingroup$ There is nothing particular about $I,J,K$ : there is in fact a sphere $\Bbb{S}^2 $ of complex structures $aI+bJ+cK $ for $a^2+b^2+c^2+0$. $\endgroup$– abxCommented Oct 7, 2023 at 15:08
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4$\begingroup$ I think you'll be able to answer this question yourself once you have the correct definitions. The actual definition is $\mathbb{C}\otimes T = T^{(1,0)}_a\oplus T^{(0,1)}_a$, where $T^{(1,0)}_a = \{ v - i\,av\ |\ v\in T\}$ and $T^{(0,1)}_a = \{ v + i\,av\ |\ v\in T\}$. $\endgroup$– Robert BryantCommented Oct 11, 2023 at 8:57
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