Strings in abstract root systems

Let $\mathrm{\Phi }$ be a subset of the simple roots of a (possibly non-reduced) abstract root system $\mathrm{\Sigma }$, and let $\lambda \in \mathrm{\Sigma }$. We define the $\mathrm{\Phi }$-string of $\lambda$ as the set of elements in $\mathrm{\Sigma }\cup \{0\}$ of the form $\lambda +{\sum }_{\alpha \in \mathrm{\Phi }}{n}_{\alpha }\alpha$, where $n_{\alpha }$ is an integer for each $\alpha \in \mathrm{\Phi }$. This notion can be regarded as some sort of generalization of the classical notion of $\alpha$-string, where $\alpha \in \mathrm{\Sigma }$.