Killing flow preserves its complex two-form

Work on a sufficiently short common local flow domain. The composition law gives ${\mathrm{\Phi }}_{-t}\circ {\mathrm{\Phi }}_t=\mathrm{id}$ and ${\mathrm{\Phi }}_t\circ {\mathrm{\Phi }}_{-t}=\mathrm{id}$ after restriction, so these smooth maps are local diffeomorphisms. Since ${\mathcal{L}}_{T}g=0$, tensor-flow invariance gives ${\mathrm{\Phi }}_t^{\ast }g=g$. Differentiating ${\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(p))={\mathrm{\Phi }}_s({\mathrm{\Phi }}_t(p))$ at $s=0$ gives $d{\mathrm{\Phi }}_t{\mathrm{\mid }}_p(T_p)=T_{{\mathrm{\Phi }}_t(p)}$. The differential has nonzero determinant and varies continuously from the identity, so these local flow maps preserve orientation.

For $v\in T_{p}M$, step 1.1 gives $({\mathrm{\Phi }}_t^{\ast }{T}^{\mathrm{♭}}{)}_p(v)=g_{{\mathrm{\Phi }}_t(p)}(T_{{\mathrm{\Phi }}_t(p)},d{\mathrm{\Phi }}_t{\mathrm{\mid }}_{p}v)=g_p(T_p,v)$. Thus ${\mathrm{\Phi }}_t^{\ast }{T}^{\mathrm{♭}}=T^{\mathrm{♭}}$. By naturality of exterior differentiation, ${\mathrm{\Phi }}_t^{\ast }F=\frac{1}{2}{\mathrm{\Phi }}_t^{\ast }d{T}^{\mathrm{♭}}=\frac{1}{2}d({\mathrm{\Phi }}_t^{\ast }{T}^{\mathrm{♭}})=F$, with the same $F$ as the Killing derivative.

An orientation-preserving local isometry preserves the inverse metric pairing on covectors, hence its determinant pairing on exterior powers. It also preserves oriented metric volume: it takes every positive orthonormal frame to a positive orthonormal frame, on which both volume forms have value one. Pull back $\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{vol}}_g$. Wedge preservation and these metric identities give $({\mathrm{\Phi }}_t^{\ast }\alpha )\wedge {\mathrm{\Phi }}_t^{\ast }({\star }_g\beta )=⟨{\mathrm{\Phi }}_t^{\ast }\alpha ,{\mathrm{\Phi }}_t^{\ast }\beta {⟩}_g{\mathrm{vol}}_g$. Since the local diffeomorphism induces an isomorphism on each exterior-power fibre, its pulled-back $\alpha$ range over all forms at each point. Uniqueness in the Hodge characterization therefore gives ${\mathrm{\Phi }}_t^{\ast }({\star }_g\beta )={\star }_g({\mathrm{\Phi }}_t^{\ast }\beta )$.

Apply step 2.2 to $\beta =F$. Step 2.1 and the componentwise complex extension give ${\mathrm{\Phi }}_t^{\ast }\mathcal{F}={\mathrm{\Phi }}_t^{\ast }F+i{\mathrm{\Phi }}_t^{\ast }({\star }_{g}F)=F+i{\star }_{g}F=\mathcal{F}$. Differentiating this identity and ${\mathrm{\Phi }}_t^{\ast }F=F$ at $t=0$ gives ${\mathcal{L}}_T\mathcal{F}=0$ and ${\mathcal{L}}_{T}F=0$ by the tensor Lie derivative definition and its complex-linear extension. All maps used were local, so no completeness of the flow is assumed.