A root pairing is a tuple \((R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\) where:

• \((R,M,N,\mathcal{L})\) is a perfect pairing,

• \(I\) is a set,

• The maps

  \begin{align*} \alpha :& \; I\to M, \quad i\mapsto \alpha _i, \\ \alpha ^\vee :& \; I\to N, \quad i\mapsto \alpha _i^\vee , \\ s:& \; I\to \operatorname {Perm}(I), \quad i\mapsto s_i, \end{align*}

  are such that

  1. \(\alpha \) and \(\alpha ^\vee \) are injective,

  2. for all \(i\in I\), \(\mathcal{L}(\alpha _i,\alpha _i^\vee )=2\),

  3. for all \(i,j\in I\),

     \begin{align*} \alpha _{s_i(j)} & =\alpha _j-\mathcal{L}(\alpha _j,\alpha _i^\vee )\alpha _i,\\ \alpha _{s_i(j)}^\vee & =\alpha _j^\vee -\mathcal{L}(\alpha _i,\alpha _j^\vee )\alpha _i^\vee . \end{align*}

Fix a root pairing \(\Phi =(R,M,N,\mathcal{L},I,\alpha ,\alpha ^\vee ,s)\). We say that \(\Phi \) is:

• crystallographic if \(\mathcal{L}(\alpha _i,\alpha _j^\vee )\in \mathbb {Z}\) for all \(i,j\in I\),

• reduced if, for all \(i,j\in I\), the roots \(\alpha _i\) and \(\alpha _j\) are linearly dependent over \(R\) if and only if \(\alpha _i=\pm \alpha _j\),

• finite if \(I\) is finite.