I have started reading about quaternionic and quaternionic Kähler manifolds. The most elegant definitions speak about the holonomy group of the manifold. However, it is possible to describe these type of manifolds using subbundles of $TM$. In particular, a manifold is called almost quaternionic if there exists a rank 3 vector subbundle $G\subset \operatorname{End}TM$ such that, locally it is spanned by 3 almost complex structures $IJ$ and $K$ verifying the quaternionic identity
$$IJK=-id_{TM}.$$
I imagine, this structures can be considered as the pre-image by the bundle trivialisations of the linear complex structures on $\mathbb H^n$, regarded as a real vector space. Hence, I have computed the complex structures on $\mathbb R^{4n}$ resulting of the operations $v\mapsto vi$, $v\mapsto vj$ and $v\mapsto vk$.
As you can see, I am considering $\mathbb H^n$ as a right module, because that is what Chevalley and others do. Then, considering a basis $\{(e_n),(f_n),(g_n),(h_n)\}_{n=1}^m$ of $\mathbb R^4$, I define $I:\mathbb R^{4n}\rightarrow \mathbb R^{4n}$ as
$$ \left\{ \begin{array}{l} Ie_n = (e_n)\cdot i = e_n i=f_n \\ If_n = (e_ni)\cdot i =-e_n \\ Ig_n = (e_nj)\cdot i = e_nji = -e_nk = -h_n\\ Ih_n=(e_nk)\cdot i = e_nki = e_nj= g_n , \end{array} \right. $$
so that $I$ is written
$$ \begin{pmatrix} 0 & -id_n & 0 & 0 \\ id_n & 0 & 0 & 0 \\ 0 & 0 & 0 & id_n \\ 0 & 0 & -id_n & 0 \\ \end{pmatrix}. $$
Similarly,
$$ J=\begin{pmatrix} 0 & 0 & -id_n & 0 \\ 0 & 0 & 0 & -id_n \\ id_n & 0 & 0 & 0 \\ 0 & id_n & 0 & 0 \\ \end{pmatrix} $$
and
$$ K=\begin{pmatrix} 0 & 0 & 0 & -id_n \\ 0 & 0 & id_n & 0\\ 0 & -id_n & 0 & 0 \\ id_n & 0 & 0 & 0 \\ \end{pmatrix}. $$
However, in this case $IJK=+id_{4n}$, while $KJI=-id_{4n}$. It seems logical because $KJI$ actin on $v$ corresponds to $v\cdot(ijk)$ and the other way round. And then it comes my question:
Why should I consider $\mathbb H$ acting on the right for the module structure but acting on the left for computing the complex structures $I,J$ and $K$? It seems me cheating.
EDIT:
This pdf is attached in a comment below this question. In page 35 it says:
More generally let $V=\mathbb H^n\cong =\mathbb R^{4n}$ be a "quaternionic vector space". We view it a sa left $\mathbb H$-module by left multiplication. $V$ has three complex structures $I,J,K$ given by left (component-wise) multiplication by $i,j,k$ respectively.
Let me emphasize the left multiplication. However, then the text says:
If we let $g\in GL_{4n}(\mathbb R)$ act on $q\in V$ as $qg^{−1}$ (where we view $q$ as a row vector), then the action of $GL_n(\mathbb H)$ commutes with the action of the complex structures, i.e. it is “quaternionic-linear”
But $I,J$ and $K$ are supposed to be in $GL_{4n}(\mathbb R)$, are not they? So I cannot understand how it is possible to consider them acting on the left first and then on the right.
P.S. As you probably have noticed, I have focused on almost quaternionic manifolds, since I am interested in the way complex structures have to be define. I speak neither the integrability of the complex structures nor the associated Kählerian forms $\omega_I,\omega_J,\omega_K$. In fact, what I really want is to understand how a(n almost) quaternionic (Kähler) structures works, trying to understand how it reduces the structure group.
