define ${\operatorname{F}}$, ${\operatorname{G}}\in {\operatorname{End}}({\wedge^2 T^*M})$ by $$\begin{aligned}
 ({\operatorname{F}}\eta)(X, Y) &= \frac{1}{2}\left(\eta(X, JY) + \eta(JX, Y)\right), \quad ({\operatorname{G}}\eta)(X, Y) = \eta(JX, JY),\end{aligned}$$ for $\eta \in {\wedge^2 T^*M}$ and $X$, $Y$ in $TM$, and let $${\tilde{{\operatorname{P}}}}= \frac{1}{2}(\operatorname{Id} + \mathrm{G}), \quad {\hat{{\operatorname{P}}}}= \frac{1}{2}(\operatorname{Id} - {\operatorname{G}}).$$ The identities $$\label{eq:gfrel}
  \quad {\operatorname{G}}^2 = \operatorname{Id}, \quad {\operatorname{F}}{\operatorname{G}}= {\operatorname{G}}{\operatorname{F}}= - {\operatorname{F}}$$ satisfied by ${\operatorname{F}}$ and ${\operatorname{G}}$ imply the following relations between ${\operatorname{F}}$, ${\tilde{{\operatorname{P}}}}$, and ${\hat{{\operatorname{P}}}}$.

\[lem:projprop\] On $M\times[0, T]$ we have the identities $$\begin{aligned}
 &{\tilde{{\operatorname{P}}}}^2 = {\tilde{{\operatorname{P}}}}, \quad {\hat{{\operatorname{P}}}}^2 = {\hat{{\operatorname{P}}}}, \quad {\hat{{\operatorname{P}}}}{\tilde{{\operatorname{P}}}}= {\tilde{{\operatorname{P}}}}{\hat{{\operatorname{P}}}}= 0, \quad {\tilde{{\operatorname{P}}}}+ {\hat{{\operatorname{P}}}}= \operatorname{Id},\\ 
 &\qquad{\tilde{{\operatorname{P}}}}^* = {\tilde{{\operatorname{P}}}}, \quad {\hat{{\operatorname{P}}}}^* = {\hat{{\operatorname{P}}}}, \quad {\operatorname{F}}^* = - {\operatorname{F}},\\
 &{\operatorname{F}}^2 = - {\hat{{\operatorname{P}}}}, \quad {\tilde{{\operatorname{P}}}}{\operatorname{F}}= {\operatorname{F}}{\tilde{{\operatorname{P}}}}= 0,
 \quad {\operatorname{F}}{\hat{{\operatorname{P}}}}= {\hat{{\operatorname{P}}}}{\operatorname{F}}= {\operatorname{F}}.\end{aligned}$$ Here $\,^*$ denotes the adjoint of the operator relative to the metric .

The first two rows of relations imply, among other things, that $$\mathcal{H} = \mathcal{H}(\tau) = \operatorname{im}{\tilde{{\operatorname{P}}}}(\tau), \quad \mathcal{K} = \mathcal{K}(\tau) = \operatorname{im} {\hat{{\operatorname{P}}}}(\tau),$$ are complementary orthogonal subbundles of ${\wedge^2 T^*M}$.

\[lem:splitting\] For any $(p, \tau) \in M\times [0, T]$, we have the $g(\tau)$-orthogonal decomposition $$\label{eq:splitting}
    \wedge^2(T_p^*M) = \mathcal{H}_p(\tau)\oplus \mathcal{K}_p(\tau)$$ and $\mathcal{H}_p(\tau)$ is a subalgebra of $\wedge^2(T_p^*M)$ isomorphic to $\mathfrak{u}(n/2)$ relative to the bracket $$\label{eq:bracket}
 [\omega, \eta]_{ij} = \omega_{ik}\eta_{kj} - \omega_{jk}\eta_{ki}$$ induced 