**(A)** In this really stylish [answer](https://mathoverflow.net/questions/220214/is-there-an-integrable-complex-structure-on-su3/220230#220230) it is shown that one can define a family of complex structures $J\_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\mathbb R}$. 


**(B)** [Wikipedia](https://en.wikipedia.org/wiki/Hypercomplex_manifold) can tell us that $SU(3)$ has a hypercomplex structure. 


I am searching a description of **(B)** using the language of **(A)**.



The Hypercomplex Structure of $SU(3)$
I'm assuming that you have a copy of Dominic Joyce's 1992 JDG article, "Compact hypercomplex and quaternionic manifolds" handy. Write $\frak{g} = \frak{su}(3)$ as a direct sum
$$
\frak{su}(3) = \frak{b}\ \oplus\ \frak{d}\ \oplus\ \frak{f}\ ,
$$
where
$$
{\frak{b}} = \left\{\begin{pmatrix}ia&0&0\\0&ia&0\\0&0&-2ia\end{pmatrix}\ \Biggl|\ a\in\mathbb{R}\ \right\}\simeq \mathbb{R}
$$
$$
{\frak{d}} = \left\{\begin{pmatrix}ip&q+ir&0\\-q+ir&-ip&0\\0&0&0\end{pmatrix}\ \Biggl|\ p,q,r\in\mathbb{R}\ \right\}\simeq{\frak{su}}(2)
$$
$$
{\frak{f}} = \left\{\begin{pmatrix}0&0&-\overline z\\0&0&-\overline w\\z&w&0\end{pmatrix}\ \Biggl|\ z,w\in\mathbb{C}\ \right\}\simeq \mathbb{C}^2\simeq\mathbb{H}.
$$
One can easily check that this decomposition satisfies all of the conditions of Lemma 4.1 on page 751 (where, since $n=1$ in this case, I'm omitting the subscripts and summation signs). Now, in Theorem 4.2, we have $k=m=0$, and we can define the complex structures $I\_1$, $I\_2$, $I\_3$ on ${\frak{su}}(3)$ satisfying $I\_1I\_2=I\_3$ as on pages 753–754. (This depends on choosing a basis of $\frak{b}$, so there is a $1$-parameter family of such choices.) By Lemma 4.3, these extend by left-invariance to integrable almost complex structures on $G=\mathrm{SU}(3)$, and so they define a left-invariant hypercomplex structure on $\mathrm{SU}(3)$.


**Addendum:** The OP's request was for a 'description of (B) using the language of (A)', and the above answer doesn't quite do that. Moreover, while thinking about the best way to honor the original request, I took another look at Joyce's paper and realized that there is a sign mistake in one of his formulae that is liable to confuse whoever tries to carry out that description (maybe the OP?) and lead to some frustration. 


First, the sign error: When Joyce goes to construct the three complex structures $I\_1$, $I\_2$, and $I\_3$ on ${\frak{f}} \subset {\frak{su}}(3)$ at the top of page 754, he gives the formula $I\_a(v) = \bigl[v,\phi\_j(i\_a)\bigr]$, but this would lead to $I\_1I\_2=-I\_3$ on $\frak{f}$ instead of the desired $I\_1I\_2=I\_3$, so his definition should instead be $I\_a(v) = \bigl[\phi\_j(i\_a),v\bigr]$, and then everything works out correctly.


The upshot is that, if we then introduce a basis $\theta\_i$ for the left-invariant $1$-forms on $\mathrm{SU}(3)$ in such a way that the canonical left-invariant form is
$$
\gamma = g^{-1}\,\mathrm{d}g 
= \begin{pmatrix} 
i(\theta\_1+\theta\_2) & i\theta\_3-\theta\_4 & \theta\_5 + i\theta\_6\\ 
i\theta\_3+\theta\_4 & i(\theta\_1-\theta\_2) & i\theta\_7+\theta\_8\\
-\theta\_5+i\theta\_6 & i\theta\_7 - \theta\_8 & -2i\theta\_1
\end{pmatrix},
$$
then the $\mathbb{H}^2$-valued left-invariant $1$-form
$$
\theta = \bigl(\theta\_1+i\theta\_2 + j\theta\_3 +k\theta\_4\,,\ 
 \theta\_5+i\theta\_6 + j\theta\_7 +k\theta\_8 \bigr)
$$
satisfies $\theta(I\_1v) = i\theta(v)$ and $\theta(I\_2v) = j\theta(v)$
for all $v\in {\frak{su}}(3)$. As a result, the $\mathbb{C}^4$-valued $1$-forms
$$
\alpha = \bigl(\ \theta\_1{+}i\theta\_2\ \ \theta\_3{+}i\theta\_4\ \ 
 \theta\_5{+}i\theta\_6\ \ \theta\_7{+}i\theta\_8\ \bigr)
=(\alpha\_1\ \alpha\_2\ \alpha\_3\ \alpha\_4)
$$
and 
$$
\beta = \bigl(\ \theta\_1{+}i\theta\_3\ \ \theta\_2{-}i\theta\_4\ \ 
 \theta\_5{+}i\theta\_7\ \ \theta\_6{-}i\theta\_8\ \bigr)
=(\beta\_1\ \beta\_2\ \beta\_3\ \beta\_4)
$$
satisfy $\alpha(I\_1v) = i\alpha(v)$ and $\beta(I\_2v) = i\beta(v)$ for all tangent vectors $v\in T\mathrm{SU}(3)$. Consequently the $\alpha\_i$ are a basis for the $(1,0)$-forms on $\mathrm{SU}(3)$ with respect to the almost-complex structure $I\_1$ while the $\beta\_i$ are a basis for the $(1,0)$-forms on $\mathrm{SU}(3)$ with respect to the almost-complex structure $I\_2$. 


Finally, using the identity $\mathrm{d}\gamma = -\gamma\wedge\gamma$ to get the formulae for $\mathrm{d}\theta\_i$, we find that 
$$
\mathrm{d}\alpha\_i\equiv 0\ \mod\ \alpha\_1,\alpha\_2,\alpha\_3,\alpha\_4
$$
while
$$
\mathrm{d}\beta\_i\equiv 0\ \mod\ \beta\_1,\beta\_2,\beta\_3,\beta\_4\,.
$$


Thus, $I\_1$ and $I\_2$ (and hence $I\_3 = I\_1I\_2=-I\_2I\_1$) are integrable complex structures on $\mathrm{SU}(3)$, and thus the triple $(I\_1,I\_2,I\_3)$ defines a hypercomplex structure on $\mathrm{SU}(3)$.


