Let $G$ be a linear algebraic group, $T$ a subtorus of $G$, and $\mathfrak g$ the Lie algebra of $G$. Then there exist characters $\chi_1, ... , \chi_t$ of $T$, and subspaces $V_i = V_{\chi_i}$ of $\mathfrak g$, such that $\mathfrak g$ is the direct sum of the $V_i$, and $\textrm{Ad }x(v) = \chi_i(x)v$ for any $x \in T, v \in V_i$. The characters $\chi_i$, excluding the trivial character if it occurs, are called the weights of $T$.
Fix a weight $\chi$, and let $g \in N_G(T)$. Then $g$ induces a character $\chi'$ of $T$ given by $\chi'(x) = \chi(gxg^{-1})$. I am trying to understand why $\chi'$ is also a weight of $T$. In other words, given that there exists a $0 \neq v \in \mathfrak g$ such that $\textrm{Ad }x(v) = \chi(x)v$ for all $x \in T$, why is it the case that there exists a $0 \neq w \in \mathfrak g$ such that $\textrm{Ad }x(w) = \chi(gxg^{-1})w$ for all $x \in T$?