Sub-coroot systems

Question

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?

Answer 1

This can fail. Consider the natural embedding $\operatorname{Sp}_4(\mathbb C) \to \operatorname{SL}_4(\mathbb C)$, where $\operatorname{Sp}_4$ is taken with respect to the symplectic form $(x, y) \mapsto x_1 y_4 + x_2 y_3 - x_3 x_2 - x_4 y_1$. (This is a map of complex Lie groups, but it carries maximal compact subgroups to maximal compact subgroups, so you can work in that setting if you prefer.) If we denote the simple (with respect to the upper-triangular Borel) roots of the diagonal torus in $\operatorname{SL}_4(\mathbb C)$ by $\alpha_1, \alpha_2, \alpha_3$, in the obvious fashion, then the restrictions of $\alpha_1$ and $\alpha_3$ to the diagonal torus in $\operatorname{Sp}_4(\mathbb C)$ are the short simple (with respect to the upper-triangular Borel) root $a$, and the restriction of $\alpha_2$ is the long simple root $b$. Then $a^\vee$ equals $\alpha_1^\vee + \alpha_3^\vee$, which is not a coroot of $\operatorname{SL}_4(\mathbb C)$.

(As discussed in comments, the original version of the question did not include the condition on centres, and so admitted any proper, maximal-rank subgroup as a counterexample. This example does satisfy the centre condition.)