In Klingenberg's Lectures on Closed Geodesics, he states a proposition that goes as follows:
Proposition: A connection $K$ on $\pi: E \rightarrow M$ defines a splitting
$TE=T_hE \oplus T_vE$
of the tangent bundle with
$T_{\xi}E=T_{\xi h}E \oplus T_{\xi v}E$
More precisely, under the canonical identification of $T_{\xi v}E$ with $E_p$, $p=\pi(\xi)$, we can write this decomposition as
$T_{\xi}E=(id-K)T_{\xi}E + KT_{\xi}E$
and then proceeds by saying that:
Proof: By looking at the local representation $K_{\phi}$ of $K$ we see that, if we identify $T_{\xi v}$ with $E_p$, i.e. if we identify
$\{(x,\xi, 0, \eta)\} \in \{(x,\xi)\}\times \mathbb{M}\times \mathbb{E}$
with
$\{(x, \eta)\} \in \{x\} \times \mathbb{E}$,
then $K_{\phi}^2=K_{\phi}$...
and I don't get it. Because, shouldn't $K_{\phi}^2=(x, \eta +2 \Gamma _{\phi}(x)(y,\xi))\neq (x, \eta + \Gamma_{\phi}(x)(y,\xi))=K_\phi$? What am I overlooking?