Let $X$ be a smooth manifold, let $T_{\mathbb C}X := TX \otimes_{\mathbb R} \mathbb C$ be its complexified tangent bundle, and let $\nabla$ be a $\mathbb C$-linear connection on $T_{\mathbb C}X$. Let $[\cdot,\cdot]$ denote the $\mathbb C$-bilinear extension of the Lie bracket of real vector fields. Define the torsion tensor
In particular, when $\nabla$ is the Chern connection on $T^{1,0}X$, the same identity holds after extending the connection and bracket complex-linearly and then restricting or projecting the resulting identity to the relevant type components.