Integer quadratic forms and extensions of subsets of linearly independent roots

We consider subsets of linearly independent roots in a certain root system $\varPhi$. Let $S'$ be such a subset, and let $S'$ be associated with any Carter diagram $\Gamma'$. The main question of the paper: what root $\gamma \in \varPhi$ can be added to $S'$ so that $S' \cup \gamma$ is also a subset of linearly independent roots? This extra root $\gamma$ is called the linkage root. The vector $\gamma^{\nabla}$ of inner products $\{(\gamma,\tau'_i)\mid \tau'_i \in S'\}$ is called the linkage label vector. Let $B_{\Gamma'}$ be the Cartan matrix associated with $\Gamma'$. It is shown that $\gamma$ is a linkage root if and only if $\mathscr{B}^{\vee}_{\Gamma'}(\gamma^{\nabla}) < 2$, where $\mathscr{B}^{\vee}_{\Gamma'}$ is a quadratic form with the matrix inverse to $B_{\Gamma'}$. The set of all linkage roots for $\Gamma'$ is called a linkage system and is denoted by $\mathscr{L}(\Gamma')$. The Cartan matrix associated with any Carter diagram $\Gamma'$ is conjugate to the Cartan matrix associated with some Dynkin diagram $\Gamma$, [St23]. The sizes of $\mathscr{L}(\Gamma')$ and $\mathscr{L}(\Gamma)$ are the same. Let $W^{\vee}$ be the Weyl group of the quadratic form $\mathscr{B}^{\vee}_{\Gamma'}$. This group acts on the linkage system and forms several orbits. The sizes and structure of orbits for linkage systems $\mathscr{L}(D_l)$ and $\mathscr{L}(D_l(a_k))$ are presented.