Given a linear map $J: T \mathbb{R}^4 \to T\mathbb{R}^4$ $$ J= \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ f_1 & f_3 & 0 & 1 \\ f_2 & f_4 & -1 & 0 \end{pmatrix} $$ Under what conditions on $\{f_j\}_{j=1}^4 \subset C^\infty(\mathbb{R}^4)$ is $J$
- an almost complex structure
- A complex structure
The first question seems okay, multiplying the matrix with itself yields $J^2 = - $id $\iff f_2 = f_3, f_1 = -f_4$. Now for the integrability condition, I know that an almost complex structure of the form $$ \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0& 0 & -1 & 0 \end{pmatrix} $$ defines a complex structure, but is this a necessary condition for integrability?