Example

(1) The trivial system of commutation factors consists of the $\varepsilon_{ij}$ such that $\varepsilon_{ij}(\alpha_i, \beta_j) = 1$ for all $i, j$, $\alpha_i \in \Delta_i$, $\beta_j \in \Delta_j$.

(2) If we take $\mathbf{A} = \mathbf{Z}$ and $\Delta_i = \mathbf{Z}$ for all $i \in I$, a system of commutation factors is obtained by taking $\varepsilon_{ij}(\alpha_i, \beta_j) = (-1)^{\alpha_i \beta_j}$. Note that this number depends only on the parities of $\alpha_i$ and $\beta_j$ and the $\varepsilon_{ij}$ can therefore be considered as commutation factors when certain of the $\Delta_i$ are equal to $\mathbf{Z}/2\mathbf{Z}$ and the others to $\mathbf{Z}$.

These two examples are the most frequent cases encountered in applications.