Let $\mathbb{M}=G/\Gamma$ be the Iwasawa manifold, namely, the quotient space of the set of all matrices of the form
$$ G= \left\{ (z_1,z_2,z_3) := \begin{pmatrix} 1 & z_1 & z_3 \\ 0 & 1 & z_2 \\ 0 & 0 & 1 \end{pmatrix} : z_1,z_2,z_3 \in \mathbb{C} \right\} $$
divided by the subgroup of $\Gamma$ consisting of all matrices whose entries are Gaussian integers, namely, $z_1,z_2,z_3 \in \mathbb{Z}[i]=\{a+b\sqrt{-1} ; a,b \in \mathbb{Z}\}$.
Note that the left action is $$ \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & z_1 & z_3 \\ 0 & 1 & z_2 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & z_1+a & z_3+a z_2+c \\ 0 & 1 & z_2+b \\ 0 & 0 & 1 \end{pmatrix}. $$
we regard $\mathbb{M}$ as a bundle by the map $p:\mathbb{M} \to T^4;(z_1,z_2,z_3) \mapsto (z_1,z_2)$.
Consider left invariant 1 forms $dz_1 ,dz_2,-dz_3-z_1dz_2$ on $\mathbb{M}$ and we also introduce 1-forms $e^i,i=1,...,6$ as follows $$dz_1=e^1+ie^2, \\dz_2=e^3+ie^4,\\-dz_3-z_1dz_2=e^5+ie^6$$
Then, the following theorem holds.
Theorem Let $J$ be any left invariant complex structure on $\mathbb{M}$. Then $p$ induces a complex structure $ J'$ such that $p:(\mathbb{M},J) \to (T^4,J')$ is holomorphic.
To prove this theorem, in a part of the proof in a paper, it says the following.( a complete proof is not given here , only special case are proven in the following).
Let $\Lambda$ denote the space of $(1,0)$ forms relative to $J$. Then
dim$(<e^1,e^2,e^3,e^4,e^5>_c \cap \Lambda) = 2$. -------(A)
If
dim$(<e^1,e^2,e^3,e^4> \cap \Lambda) =2$,
then $J( <e^1,e^2,e^3,e^4> )\subset <e^1,e^2,e^3,e^4>$ , -----(B)
as required.
I know the space $<e^1,e^2,e^3,e^4>$ is the cotangent space of $T^4$, so the $J( <e^1,e^2,e^3,e^4> )\subset <e^1,e^2,e^3,e^4>$ indicates that the restriction $J':=J|_{<e^1,e^2,e^3,e^4>}$ is well-defined. But I cannot understand why the above two sentences (A),(B) hold.
Note that $<e^1,e^2,e^3,e^4,e^5>$ denotes a linear space spanned by the specified vectors $e_1,e_2,e_3,e_4$.
I am not sure what $<e^1,e^2,e^3,e^4,e^5>_c$ means.
Ref.
http://www.kurims.kyoto-u.ac.jp/EMIS/journals/AG/4-2/4_165.pdf
page 167 , Theorem 1.1
