\p^\+\wedge A^-& F^+-\ve\,\p^\+\wedge\p
\end{pmatrix} =0\ ,$$ where $F^\pm = \diff A^\pm + A^\pm\wedge A^\pm$. From (\[4.15\]) we get $$\label{4.16}
F^- = \ve\,\p\wedge\p^\+ = - \frac{\ve}{4\La^2}
\begin{pmatrix} \th^1\wedge\th^{\1} - \th^2\wedge\th^{\2}& 2\th^{\1}
\wedge\th^2\\
 -2\th^1\wedge\th^{\2}&-\th^1\wedge\th^{\1} + \th^2\wedge\th^{\2}
\end{pmatrix}\ ,$$ $$\label{4.17}
F^+ = \ve\,\p^\+\wedge\p = - \frac{\ve}{4\La^2}
\begin{pmatrix} \th^1\wedge\th^{\1} + \th^2\wedge\th^{\2}&
2\th^{\1}\wedge\th^{\2}\\
 -2\th^1\wedge\th^{2}&-\th^1\wedge\th^{\1} - \th^2\wedge\th^{\2}
\end{pmatrix}\ .$$ One can easily see that $*F^{\pm}=\pm F^{\pm}$, i.e. $F^+$ and $F^-$ are self-dual (SD) and anti-self-dual (ASD) gauge fields on a rank-2 complex vector bundle $E\to M^4$, respectively. They can be identified with SD and ASD parts of the Riemann tensor of the metric $\diff s^2=\th^1\th^{\1}+ \th^2\th^{\2}$.

[**Twistor manifolds of $S^4$ and $B_4$ as coset spaces**]{}. Let us consider the Hopf bundle $$\label{4.18}
S^3\to S^2$$ over the Riemann sphere $S^2\cong{\mathbb C}P^1$ and the one-monopole connection $a$ on the bundle (\[4.18\]) having in the local coordinate $\zeta\in{\mathbb C}P^1$ the form $$\label{4.19}
a=\frac{1}{2(1+\zeta\bar\zeta)}\, (\bar\zeta\,\diff\zeta -\zeta\,
\diff\bar\zeta)\ .$$ Consider a local section of the bundle (\[4.18\]) given by the matrix $$\label{4.20}
g=\frac{1}{(1+\zeta\bar\zeta)^{\frac{1}{2}}}\,
\begin{pmatrix}1&-\bar\zeta\\ \zeta & 1\end{pmatrix}
\in {\rm SU}(2)\cong S^3$$ and introduce the $su(2)$-valued one-form (flat connection) $$\label{4.21}
g^{-1}\diff g=:
\begin{pmatrix}a&-\frac{1}{2R}\th^{\3}\\
\frac{1}{2R}\th^{3} & -a\end{pmatrix}\ ,$$ where $$\label{4.22}
\th^3=\frac{2R\diff\zeta}{1+\zeta\bar\zeta}\und
\th^{\3}=\frac{2R\diff\bar\zeta}{1+\zeta\bar\zeta}$$ are the forms of type (1,0) and (0,1) on ${\mathbb C}P^1$, 