I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on symplectic manifolds.

Equation 3.27 on page 47 reads as \begin{equation} H_V=\mathcal{L}_V-[\iota_V,\nabla]=\iota_V\theta=V^{\mu}\theta_{\mu}, \end{equation} where $H_V$ is the moment map evaluated on an element of the Lie algebra of $G$, with associated vector field $V$, and $\theta$ is the connection 1-form on the line bundle where $\nabla=d+\theta$. I am not completely sure how the above equality follows, and it seems to me that there is a sign error. Here is what I have: \begin{equation} \begin{aligned} H_V&=\{d,\iota_V\}-[\iota_V,\nabla]\\ &=\{d,\iota_V\}-[\iota_V,d+\theta]\\ &=d\iota_V+\nabla\iota_V-\iota_V\theta. \end{aligned} \end{equation}

I am not sure how the first two terms vanish, and why the last term has a minus sign. Any help would be much appreciated.