by $g(\tau)$.

As we have noted, the first assertion follows from the first two rows of Lemma \[lem:projprop\]. For the second assertion, let $\mathfrak{s}_p(\tau)\cong \mathfrak{so}(n)$ denote the Lie subalgebra of endomorphisms of $T_pM$ (under commutation) which are skew-symmetric relative to $g(p, \tau)$. The endomorphisms in $\mathfrak{s}_p(\tau)$ which further commute with $J = J(p, \tau)$ comprise a subalgebra $\mathfrak{u}_p(\tau)$ of $\mathfrak{s}_p(\tau)$ isomorphic to $\mathfrak{u}(n/2)$. The map $\Phi_{p, \tau}: \mathfrak{s}_p(\tau) \to \wedge^2(T^*_pM)$ given by $\Phi_{p, \tau}(A)(X, Y) =  g(X, AY)$ is an isomorphism relative to the bracket on $\wedge^2(T^*_pM)$, and the image of $\mathfrak{u}_p(\tau)$ under $\Phi$, consisting of those two-forms $\eta$ for which $\eta(X, Y) = \eta(JX, JX)$ for all $X$, $Y\in T_pM$, is precisely $\mathcal{H}_p(\tau)$.

Relative to $\wedge_{{\mathbb{C}}}^2T^*M = {\wedge^2 T^*M}\otimes_{{\mathbb{R}}} {\mathbb{C}}$, we have $${\operatorname{P}}^{(2, 0)} = \frac{1}{2}\left({\hat{{\operatorname{P}}}}- \sqrt{-1}{\operatorname{F}}\right),\quad
 {\operatorname{P}}^{(0, 2)} = \frac{1}{2}\left({\hat{{\operatorname{P}}}}+ \sqrt{-1}{\operatorname{F}}\right),\quad
 {\operatorname{P}}^{(1, 1)} = {\tilde{{\operatorname{P}}}},$$ where ${\operatorname{P}}^{(2, 0)}$, ${\operatorname{P}}^{(0, 2)}$, ${\operatorname{P}}^{(1, 1)}$ are the projections of $\wedge_{{\mathbb{C}}}^2T^*M$ onto $\wedge^{2, 0}M$, $\wedge^{0, 2}M$, and $\wedge^{1, 1}M$, respectively.

The decomposition of the curvature operator
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Let us use $R$ to denote the $(4, 0)$ curvature tensor and ${\operatorname{R}}:{\wedge^2 T^*M}\to {\wedge^2 T^*M}$ to denote the curvature operator of $g(\tau)$. We adopt the convention that $${\operatorname{R}}(\eta)_{ij} = -R_{ijab}\eta_{ab}, \quad \eta\in {\wedge^2 T^*M}.$$ 