I am given a Hermitian connection $\nabla$ of a Hermitian vector bundle $\pi:E\rightarrow M$. In other words i have a Hermitian product $h_p$ on $E_p$, such that the Riemannian metric is given via $g=Re(h)$, on a vector bundle with a $J\in End(E_p)$, such that $J^2=-1$ and a connection that is metric with $g$.
Trying to understand the structure i tried to proof two remarks given in the lecture: \begin{align} \text{It follows from the definition that: }&\quad h(e,f)=g(e,f)+ig(e,J(f))\\ \text{Regarding the curvature it follows: }&\quad h(F(e),f)=-h(e,F(f)) \end{align}
I have no idea for the first remark, but the for the second remark i have the following \begin{align} h(e,F(f))=&g(e,F(f))+i g(e,J(F(f)))\\ &\text{since $\nabla J=0$, the curvature commutes $J(F(f))=F(J(f))$}\\ =&g(e,F(f))+i g(e,F(J(f)))\\ &\text{skew symmetry of g wrt. the curvature gives us}\\ =&-g(F(e),f)-i g(F(e),J(f))\\ =& -h(F(e),f) \end{align} I would appreciate ideas for the first remark and verification for my proof, since I am new to the concept.
The second one seems correct to me, while for the first one you could use the properties of the hermitian product and follow the directions (stop reading the steps as soon as you figure it out):
\begin{align*} \mathrm{Im}(h(e,f))&=\frac{h(e,f)-\overline{h(e,f)}}{2i}\\ &=-i\frac{h(e,f)-h(f,e)}{2}\\ &=\frac{-i\cdot h(e,f)+i\cdot h(f,e)}{2}\\ &=\frac{h(e,J(f))+h(J(f),e)}{2}\\ &=\frac{h(e,J(f))+\overline{h(e,J(f))}}{2}\\ &=\mathrm{Re}(h(e,J(f)))\\ &=g(e,J(f)). \end{align*}