[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ]

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

Assume now that $K'$ is another compact Lie group that admits a homomorphic embedding $i:K\hookrightarrow K'$ verifying the condition that $i(Z_K)\subset Z_{K'}$, where $Z_K, Z_{K'}$ are the centers of $K$ and $K'$, respectively. 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 $$ \Lambda = {\mathfrak t} \cap \Lambda' \quad ?$$

where $\Lambda' \subset {\mathfrak t}'$ is tge coroot lattice of $(K',T')$.

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