group on ${{\cal T}}(S^4)=\,$Sp(2)/Sp(1)$\times$U(1) $\cong$ SU(4)/U(3) $\cong {\mathbb C}P^3$ [@Hitchin]. Similarly, for $\ve =-1$ the structure equations (\[4.35\]) define on ${{\cal T}}(B_4)=\,$Sp(1,1)/Sp(1)$\times$U(1) the Levi-Civita connection for the metric (\[4.36\]) with the holonomy group U(2,1).

Kähler geometry on twistor spaces of ${\mathbb C}P^2$ and ${\mathbb C}B_2$
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[**Manifolds ${\mathbb C}P^2$ and ${\mathbb C}B_2$ as coset spaces**]{}. We introduce ${\mathbb C}P^2$ and ${\mathbb C}B_2$ as coset spaces $$\label{5.1}
{\mathbb C}P^2 = {\rm SU(3)/S(U(1)}{\times}{\rm U(2))}\und
{\mathbb C}B_2 = {\rm SU(1,2)/S(U(1)}{\times}{\rm U(2))}$$ and denote both of them as $M^4\cong {\mathbb C}P^2$ or ${\mathbb C}B_2$ with local complex coordinates $y^{\a}$, $\a =1,2$. Consider now the principal bundles $$\label{5.2}
{\rm SU(3)}\to{\mathbb C}P^2$$ and $$\label{5.3}
{\rm SU(1,2)}\to{\mathbb C}B_2\ ,$$ both having the structure group S(U(1)${\times}$U(2))$\cong$U(1)${\times}$SU(2). Local sections of the fibrations (\[5.2\]) and (\[5.3\]) are given by 3${\times}$3 matrices $$\label{5.4}
V=\g^{-1}\begin{pmatrix}1&-\ve T^{\+}\\T&W\end{pmatrix}\in
\left\{\begin{array}{l}{\rm SU}(3)\quad {\rm for}\quad \ve =1\\
{\rm SU}(1,2)\quad {\rm for}\quad \ve =-1\end{array}\right.$$ where $$\label{5.5}
T:=\begin{pmatrix}{\bar{y}}^{\2}\\y^1\end{pmatrix} ,\quad
W:=\g\cdot{\bf 1}_2 - \frac{\ve}{\g +1}\, TT^{\+}\und
\g = (1+\ve T^{\+}T)^{\sfrac{1}{2}}=(1+\ve y^{\a}{\bar{y}}^{{\bar{\alpha}}})^{\sfrac{1}{2}}>0$$ obey $$\label{5.6}
 W^{\+}=W\ ,\quad WT=T\und W^2=\g^2\cdot{\bf 1}_2-\ve TT^{\+}$$ and therefore $$\label{5.7}
 V^{\+}\eta\,V=V\,\eta\,V^{\+}=\eta{{\qquad{\rm with}\qquad}}\eta = {\rm diag} (1, \ve , \ve )\ .$$ Matrices (\[5.4\]) with $\ve=\pm 1$ are representative elements for cosets (\[5.2\]) and (\[5.3\]) encoding information about their 