Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$.

Assume now that we have a homomorphic embedding $i:K\hookrightarrow K'$ (with $K'$ another compact Lie group) and let $T'$ be a maximal torus of $K'$ such that $i(T)\subset T'$. This gives a vector space inclusion ${\mathfrak t} \subset {\mathfrak t}'$, into the Lie algebra of $T'$.

Question: Is it true that $$ \Psi = {\mathfrak t} \cap \Psi' \quad ?$$

where $\Psi' \subset {\mathfrak t}'$ are the coroots of $(K',T')$.

If this is not always true, are there some simple conditions under which it becomes true?