Given a symplectic vector space $(V,\omega)$, consider the space $\mathcal{J}(V,\omega)$ of compatible complex structures on $V$. We can then regard $\mathcal{J}(V,\omega)$ as a symplectic manifold where each tangent space is given by $$ T_J\mathcal{J}(V,\omega) \;\; =\;\; \{X \in \text{End}(V) \; | \; JX + XJ = 0, \; \omega(X\cdot, \cdot) + \omega(\cdot, X\cdot) = 0\}, $$
the symplectic form is given by $$ \Omega_J(X,Y) \;\; =\;\; \frac{1}{2}\text{Tr}(XJY). $$
This symplectic form comes equipped with its own natural complex structure where at each tangent space $T_J\mathcal{J}(V,\omega)$ we take the complex structure to be $X \to -JX$. This then induces the compatible Riemannian metric $$ \langle X,Y\rangle \;\; =\;\; \frac{1}{2}\text{Tr}(XY). $$
I now would like to derive the Levi-Civita connection for this metric. I've been given the claim that $$ \nabla_tX(t) \;\; =\;\; \partial_t X(t) - \frac{1}{2}J(t)\left [X(t)\dot{J}(t) + \dot{J}(t)X(t)\right ] $$
where $J(t)$ is a curve in $\mathcal{J}(V,\omega)$ and $X(t) \in T_{J(t)}\mathcal{J}(V,\omega)$ is a smoothly varying vector field along this curve, and $\nabla_t$ is just short-hand for $\nabla_{\dot{J}(t)}$.
My main idea was to start by taking another vector field $Y(t)$ and try to compute $$ \dot{J}(t)\langle X(t), Y(t)\rangle \;\; =\;\; \left \langle \nabla_tX(t), Y(t)\right \rangle + \langle X(t), \nabla_tY(t)\rangle $$
and then try to exploit natural properties of the metric. Alternatively, I've thought about the general formula: $$ \langle\nabla_XY,Z\rangle \;\; =\;\; \frac{1}{2} \left (X\langle Y,Z\rangle + Y\langle Z,X\rangle - Z\langle X,Y\rangle - \langle Y, [X,Z]\rangle - \langle Z,[Y,X]\rangle + \langle X, [Z,Y]\rangle \right ). $$
This problem feels a little confusing to me since the metric itself seems flat. Would appreciate general suggestions, I'd like the satisfaction of deriving this on my own with some guidelines.