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[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 $$ \Lambda = {\mathfrak t} \cap \Lambda'$$

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

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

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As for your other question, I will work with complex Lie groups, but you can pass to maximal compact subgroups if you prefer.

Consider the natural embedding $\operatorname{SO}_4(\mathbb C) \to \operatorname{SL}_4(\mathbb C)$, where $\operatorname{SO}_4$ is taken with respect to the quadratic form $(x_1, x_2, x_3, x_4) \mapsto x_1 x_4 + x_2 x_3$. 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 common restriction of $\alpha_1$ and $\alpha_3$ to the diagonal torus in $\operatorname{SO}_4(\mathbb C)$ is a simple (with respect to the upper-triangular Borel) root $a$, and the common restriction of $\alpha_1 + \alpha_2$ and $\alpha_2 + \alpha_3$ is a simple root $b$. Then $a^\vee$ equals $\alpha_1^\vee + \alpha_3^\vee$, $b^\vee$ equals $\alpha_1^\vee + 2\alpha_2^\vee + \alpha_3^\vee$, and $\Lambda$ equals $\mathbb Z a^\vee + \mathbb Z b^\vee$, but $(\Lambda' \cap \mathfrak t)/\Lambda$ has order $2$, generated by the image of $\alpha_2^\vee$.

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    $\begingroup$ Re, in both of my examples, $K_{\mathbb C}$ is the identity component of the fixed-point group of a so-called quasi-semisimple automorphism of $K'_{\mathbb C}$ (one that stabilises both a Borel, in my case the upper-triangular Borel, and a maximal torus in that Borel, in my case the diagonal torus). From that point of view, what's going wrong in this case is the failure of the automorphism to preserve a pinning. … $\endgroup$
    – LSpice
    Commented Feb 19, 2023 at 20:09
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    $\begingroup$ … Even for a pinned automorphism, you run into trouble when your root system has a component of type $\mathsf A_{2n}$ on which your automorphism acts non-trivially. (That is, $\operatorname{SO}_5 \subseteq \operatorname{SL}_5$ exhibits the same bad behaviour I've described here.) Otherwise, for fixed points of pinned automorphisms, everything is fine; but I'm not sure if that special case is general enough to be of interest to you. $\endgroup$
    – LSpice
    Commented Feb 19, 2023 at 20:11
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    $\begingroup$ This 1 2 is described in Adler and Lansky - Root data with group actions, though related discussions have appeared elsewhere, e.g., in papers of Digne and Michel, such as, if I remember correctly, Quasi-semisimple elements. $\endgroup$
    – LSpice
    Commented Feb 19, 2023 at 20:15
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    $\begingroup$ In terms specifically of the fundamental group, if $K$ is simply connected, then $\Lambda$ is the natural integral form $X_*(T) = \operatorname{Hom}(X^*(T), \mathbb Z)$ of $\operatorname{Hom}(X^*(T), \mathbb C) = \mathfrak t_{\mathbb C}$, so at least $\Lambda' \cap \mathfrak t$ is contained in $\Lambda$ (in general, without any assumptions on how $K$ and $K'$ arise); and, if $K'$ is simply connected, then we in fact have equality. $\endgroup$
    – LSpice
    Commented Feb 19, 2023 at 20:35
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    $\begingroup$ Re, I should have said more generally that, if $K'$ is simply connected (whether or not $K$ is), then $\Lambda' \cap \mathfrak t$ contains $\Lambda$ (again, without any assumptions on how $K$ and $K'$ arise). $\endgroup$
    – LSpice
    Commented Feb 19, 2023 at 21:14

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