A}}G + G^{-1}\diff G =:
\begin{pmatrix} {\hat{A}}^-&-\ve\hat\p\\ \hat\p^\+ & {\hat{A}}^+\end{pmatrix}\ ,$$ where $$\label{4.30}
\hat\p = \p g=:-\frac{\im}{2\La}\begin{pmatrix}\o^2&\o^{\1}\\
\o^1& -\o^{\2}\end{pmatrix}\ ,\quad
{\hat{A}}^-=A^-=\begin{pmatrix}\a_-&-{\bar{\beta}}_-\\
\b_-& -\a_-\end{pmatrix}\ ,\quad
{\hat{A}}^+=:\begin{pmatrix}\hat\a_+&-\frac{1}{2R}\,\o^{\3}\\
\frac{1}{2R}\,\o^{3}& -\hat\a_+\end{pmatrix}\ ,$$ with $\a_-$, $\b_-$ given in (\[4.12\]) and $$\label{4.31}
\hat\a_+:=\frac{1}{1+\zeta{\bar{\zeta}}}\left\{(1-\zeta{\bar{\zeta}})\,\a_+ + {\bar{\zeta}}\b_+
-\zeta{\bar{\beta}}_+ +\frac{1}{2}({\bar{\zeta}}\diff\zeta - \zeta\diff{\bar{\zeta}})\right\}\ ,$$ $$\label{4.32}
\o^1:=\frac{1}{(1+\zeta{\bar{\zeta}})^{\frac{1}{2}}}\, (\th^1-\zeta\th^{\2})\ ,
\quad
\o^2:=\frac{1}{(1+\zeta{\bar{\zeta}})^{\frac{1}{2}}}\, (\th^2+\zeta\th^{\1})\ ,$$ $$\label{4.33}
\o^3:=\frac{2R}{(1+\zeta{\bar{\zeta}})^{\frac{1}{2}}}\, (\diff\zeta +\b_+ -
2\zeta\a_+ +\zeta^2{\bar{\beta}}_+)\ .$$ Note that forms (\[4.32\]) and (\[4.33\]) define on ${{\cal T}}(M^4)$ an integrable almost complex structure [@AHS] ${{\cal J}}={{\cal J}}_+$ such that $$\label{4.34}
{{\cal J}}\,\o^i = \im\,\o^i$$ with $i=1,2,3$. In other words, $\o^i$’s are (1,0)-forms w.r.t. ${{\cal J}}$.

From flatness of the connection (\[4.29\]), $\diff\hat{{\cal A}}+
\hat{{\cal A}}\wedge\hat{{\cal A}}=0$, we obtain the equations $$\label{4.35}
\diff\begin{pmatrix}\o^1\\ \o^2\\ \o^3\end{pmatrix}=
\begin{pmatrix}\hat\a_++\a_-&-\b_-&\sfrac{1}{2R}\o^{\2}\\
{\bar{\beta}}_-&\hat\a_+-\a_-&-\sfrac{1}{2R}\o^{\1}\\
-\sfrac{\ve R}{2\La^2}\,\o^2& \sfrac{\ve R}{2\La^2}\,\o^1& 2\hat\a_+
\end{pmatrix}\wedge\begin{pmatrix}\o^1\\ \o^2\\ \o^3\end{pmatrix}$$ defining the connection on ${{\cal T}}(M^4)$ with $M^4=S^4$ or $B_4$. For both cases the metric on ${{\cal T}}(M^4)$ has the form $$\label{4.36}
\diff s^2_{\ve} = \o^1\o^{\1} +\o^2\o^{\2} + \ve\o^3\o^{\3}$$ and the almost Kähler 2-form $\o$ reads[^7] $$\label{4.37}
\o_{\ve} = \sfrac{\im}{2}\,(\o^1\wedge\o^{\1} +\o^2\wedge\o^{\2} +
\ve\o^3\wedge\o^{\3})$$ From (\[4.35\]) one obtains that the 2-form (\[4.37\]) is Kähler, i.e. $\diff\o_{\ve}=0$, if and only if $R^2=\La^2$. In this case (\[4.35\]) defines for $\ve =1$ the Levi-Civita connection with U(3) holonomy 