Let $T$ be a fixed maximal torus of a compact Lie group $G$, and let the Weyl group $W=W(G,T)$ act on $\mathfrak t$, the Lie algebra of $T$, by fixing an identification of $\mathfrak t$ and its dual.
Suppose now that $G'$ is another compact Lie group and we have homomorphic embedding $i:G \hookrightarrow G'$ such that $i(T)\subset T'$, for $T'$ a maximal torus of $G'$, with $\dim T'>\dim T$, and that $i(Z_G)\subset Z_{G'}$, where $Z_G, Z_{G'}$ are the centers of $G$ and $G'$, respectively. So, we have a vector space inclusion ${\mathfrak t} \subset {\mathfrak t}'$, into the Lie algebra of $T'$.
Question: Is it true that the intersection of ${\mathfrak t}$ with a Weyl chamber for the Weyl group $W'$ of $(G',T')$ is precisely a Weyl chamber for $W$, the Weyl group of $(G,T)$ ?
If this is not always true, are there some simple conditions (such as semi-simplicity or simply-connectedness, etc) under which it becomes true?