Equivalent definitions of almost quaternionic structures

Question

I came across this thread today when I was reading about almost quaternionic structures. I was wondering if there exists an argument similar to the answer to the thread above that can show that the $GL(n,\mathbb{H})\cdot \mathbb{H}^\times$-structure definitions of an almost quaternionic structure and the definition by a pair of almost complex structures $J, K : TM\to TM$ such that $J\circ K+K\circ J=0$ (in nLabs or Sommese).

I failed to find some similar condition that a subset of frames must satisfy which leads to a principal $GL(n,\mathbb{H})\cdot \mathbb{H}^\times$-subbundle of $\mathcal F(M)$. In all the text I found, this part of equivalence is either not mentioned as some authors (like Sternberg) focus on one of the definitions, or is considered as "obvious or easy" to the readers.

Answer 1

AS far as I understand, the second "definition" that you give is not equivalent to the definition as a $G$-structure. (This is visible from the text in nLab, which I don't find very clear however.) The standard definition in direct terms is via a rank 3 subbundle $Q\subset L(TM,TM)$ which has the property that, locally around each point, it can be spanned by $J$, $K$, and $J\circ K$ for two anti-commuting almost complex structures $J$ and $K$. However, $J$ and $K$ themselves are not part of the data (and there is an $SO(3)$-freedom in their choice in each point). Indeed if you require global almost complex structures $J$ and $K$, then you get to a hypercomplex structure which corresponds to the structure group $GL(n,\mathbb H)$.

The source of the difficulty is that quaternionic scalar multiplications are not quaternionically linear maps, since the quaternions are non-commutative, and they do move the standard quaternions $i$, $j$, $k$.

For the correct definition via $Q$, there is a similar description as in the question you link to. You form the linear frame bundle of $M$ using $\mathbb H^n$ as $\mathbb R^{4n}$. Then you can either define the reduction as consiting of all linear isomorphisms $\phi:\mathbb H^n\to T_xM$ such that for each $A\in Q_x$ the map $\phi^{-1}\circ A\circ\phi\in L(\mathbb H^n,\mathbb H^n)$ is given by scalar multiplication by some imaginary quaternion. Alternatively, you take distinguished elements $J(x), K(x)\in Q_x$, use them to make $T_xM$ into a quaternionic vector space and then consider the isomorphisms $\phi$ that can be written as a quaternionic scalar multiplication followed by a quaternionically linear map.