= Solution
The <extension group> $\operatorname{Ext}^1_Q(V,W)$ consists of equivalence classes of <short exact sequences> $0\to W\to E\to V\to0$, with the zero class represented by a split sequence and addition given by the <Baer sum>. The <extension complex of quiver representations> gives
$$
0\to\operatorname{Hom}_Q(V,W)\to\bigoplus_i\operatorname{Hom}_k(V_i,W_i)\xrightarrow{\gamma_{V,W}}\bigoplus_{\rho:i\to j}\operatorname{Hom}_k(V_i,W_j)\to\operatorname{Ext}^1_Q(V,W)\to0.
$$
The printed map has $\gamma(u)_\rho=u_jf_\rho-g_\rho u_i$, so its <kernel> is $\operatorname{Hom}_Q(V,W)$ and its <cokernel> is $\operatorname{Ext}^1_Q(V,W)$. Reversing the overall differential sign changes neither identification.
For dimension vectors $\mathbf v,\mathbf w$, the <Ringel form> is
$$
\boxed{\langle\mathbf v,\mathbf w\rangle_Q=\sum_iv_iw_i-\sum_{\rho:i\to j}v_iw_j=\dim\operatorname{Hom}_Q(V,W)-\dim\operatorname{Ext}^1_Q(V,W)}.
$$
The first expression makes its dependence only on the dimension vectors explicit.
For the one-loop representation $V=k$ with loop scalar $\lambda$, $\gamma(u)=u\lambda-\lambda u=0$ on $k$. Both cochain spaces have dimension one, so \b[$\dim\operatorname{Ext}^1_Q(V,V)=1$], for every $\lambda$. Concretely, a self-extension has loop matrix $\left(\begin{smallmatrix}\lambda&c\\0&\lambda\end{smallmatrix}\right)$, with $c$ the extension parameter.
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