Given a root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\), a set \(\Delta \subset I\) is called a base of \(\Phi \) if:

1. the sets \(\{ \alpha _i:i\in \Delta \} \) and \(\{ \alpha _i^\vee :i\in \Delta \} \) are \(R\)-linearly independent, and

2. for every \(x\in I\), there exist \(k,m\in \mathbb {Z}_{{\gt}0}\), \(a_1,\ldots ,a_k,b_1,\ldots ,b_m\in \Delta \), and \(\varepsilon ,\delta \in \{ \pm 1\} \) such that

   \[ \alpha _x=\varepsilon \sum _{r=1}^k\alpha _{a_r} \quad \text{and}\quad \alpha _x^\vee =\delta \sum _{r=1}^m\alpha _{b_r}^\vee . \]

A base \(\Delta \) is said to be irreducible if there exist \(k\in \mathbb {Z}_{{\gt}0}\) and \(a_1,\ldots ,a_k\in \Delta \) such that:

1. \(\{ a_1,\ldots ,a_k\} =\Delta \), and

2. \(\mathcal{L}(\alpha _{a_r},\alpha _{a_{r+1}}^\vee )\ne 0\) for every \(1\le r{\lt}k\).

By definition, a root pairing admitting an irreducible base is called irreducible.

Fix a finite, crystallographic, reduced and irreducible root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\) and a base \(\Delta \subset I\). The pair \((\Phi ,\Delta )\) is said to be of:

1. type \(A\) if there is a sequence \((i_1,\ldots ,i_n)\), where \(n=|\Delta |\), of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n; \]

2. type \(B\) if there is a sequence \((i_1,\ldots ,i_n)\) of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n-1, \qquad \mathcal{L}(\alpha _{i_n},\alpha _{i_{n-1}}^\vee )=-2; \]

3. type \(C\) if there is a sequence \((i_1,\ldots ,i_n)\) of distinct elements of \(\Delta \) such that

   \[ \mathcal{L}(\alpha _{i_r},\alpha _{i_{r+1}}^\vee )=-1 \quad \text{for all }1\le r{\lt}n-1, \qquad \mathcal{L}(\alpha _{i_{n-1}},\alpha _{i_n}^\vee )=-2; \]

4. type \(D\) if there exist distinct elements \(d_1,d_2,t\in \Delta \) such that

   \[ \mathcal{L}(\alpha _{d_a},\alpha _t^\vee )=-1 \quad (a=1,2), \]

   and \(\Delta \setminus \{ d_1,d_2\} \) is of type \(A\);

5. type \(E\) if there exist distinct elements \(d_1,d_2,t\in \Delta \) such that

   \[ \mathcal{L}(\alpha _{d_1},\alpha _{d_2}^\vee )=-1, \qquad \mathcal{L}(\alpha _{d_2},\alpha _t^\vee )=-1, \]

   and \(\Delta \setminus \{ d_1,d_2\} \) is of type \(A\).