= Solution
With the normalization used here, $N=2N_J$, where $N_J$ is the standard <Nijenhuis tensor>. First check that it is a <tensor>: the <Lie bracket> identity $[fU,V]=f[U,V]-V(f)U$ gives
$$
\begin{aligned}
N_J(fU,V)-fN_J(U,V)
&=-(JV)(f)JU+V(f)U+(JV)(f)JU+V(f)J^2U\\
&=0.
\end{aligned}
$$
Skew symmetry gives the same $C^\infty$-linearity in the second input. In coordinates from a <holomorphic atlas>, the real coordinate <vector fields> $\partial_{x_j},\partial_{y_j}$ commute, and $J$ sends each to another such field with a constant sign. Every <Lie bracket> in $N_J$ therefore vanishes on these coordinate fields. Tensoriality then gives \b[$\boxed{N=0}$] on all smooth <vector fields>. The factor two in this paper does not change this vanishing conclusion.
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