Lemma 1

For all $i$, $\sum_{j \neq i} q_{ij}^2 < 1$.

Let $J$ be the set of $j \in I$ such that $q_{ij} \neq 0$, in other words such that $\{i, j\}$ is an edge of $X$. If $j, j' \in J$ and $j \neq j'$, $\{j, j'\}$ is not an edge (otherwise $i, j, j'$ would form a circuit), so $(e_j|e_{j'}) = 0$. Let $F = \sum_{j \in J} \mathbf{R} e_j$. Then $(e_j)_{j \in J}$ is an orthonormal basis of $F$. The distance $d$ from $e_i$ to $F$ is given by $d^2 = 1 - \sum_{j \in J} (e_i|e_j)^2 = 1 - \sum_{j \in J} q_{ij}^2 = 1 - \sum_{j \neq i} q_{ij}^2$, hence the lemma.