Let $(M,g)$ be a closed Riemannian manifold of dimension $4$. We denote its twistor space, the space of almost complex structures on the tangent bundle $TM$ by $Z\xrightarrow{\pi}M$. At any point the fibre is isomorphic to $S^2$. Now this can be seen as $S(\Lambda_+^2M),$i.e., the sphere bundle of the self-dual two forms on $M$. Self dual means $*\omega=\omega$.
$1.$ I want to know how this description works; i.e., given a self dual two form at a point $m\in M,$ how do we get an almost complex structure on $T_m M?$
$2.$ Is it possible to induce a metric on $Z$ from $M$, if yes how?
$3.$ Once we put a metric on $Z$ and take the corresponding Levi-Civita connection we get a splitting of the tangent bundle in terms of horizontal and vertical vectors: \begin{equation*} TZ=TV\oplus TH\cong TV\oplus \pi^*(TM) \end{equation*} Does this give a reduction of the structure group $SO(6)$ to $SO(4)\times SO(2)?$