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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 $K'$ 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 $K'$. 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 1: 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?

[Edited after the first question was answered negatively]

Question 2: Let $\Lambda$ and $\Lambda '$ be the coroot lattices of $\Psi$ and $\Psi '$. Is it true that

$$ \Lambda = {\mathfrak t} \cap \Lambda' \quad ?$$

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 $K'$ 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 $$ \Psi = {\mathfrak t} \cap \Psi' \quad ?$$

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?

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 $K'$ 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 $K'$. 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?

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 $K'$ 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 $K'$. 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 1: 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?

[Edited after the first question was answered negatively]

Question 2: Let $\Lambda$ and $\Lambda '$ be the coroot lattices of $\Psi$ and $\Psi '$. Is it true that

$$ \Lambda = {\mathfrak t} \cap \Lambda' \quad ?$$

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?

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 $K'$ 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 $K'$. 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?