I am interested in a description of characters of parabolic subgroups in terms of simple roots. More precisely, suppose $G$ is a semisimple algebraic group. Fix a maximal torus, for which $\Phi$ is the system of roots, and a system of simple roots $\{\alpha_1,\ldots, \alpha_l\}$. Let $P$ be the parabolic subgroup corresponding to a subset $I\subset \{\alpha_1,\ldots, \alpha_l\}$. (Thus, the Lie algebra of $P$ is $\mathfrak{t}\oplus \bigoplus_{\alpha\in \Phi^+} \mathfrak{g}_{\alpha} \oplus \bigoplus_{\beta \text{ is a }\mathbb{Z}_{\ge 0} \text{ linear combination of }\alpha_i\in I}\mathfrak{g}_{-\beta}$.)
Claim: The characters of $P$ are precisely those of the form $\sum_{ j\not \in I} c_j \alpha_j$ for $c_j\in \mathbb{Z}$.
Is this claim of mine true? If it is, can someone outline why or point to an English reference? I am aware of Group of characters for parabolic subgroups, but I don't see directly if this answers my question.
Thanks.