TITLE: Maximally commuting structure inside a nonabelian Lie group [SU($N$), SO($N$)]
QUESTION [2 upvotes]: Consider a non-abelian compact Lie group $G$. Pick up any generator $T$ of Lie algebra of this $G$. This generator $T$ generates a U(1) subgroup inside $G$. 
Question: What is the structure (group?) inside this $G$ that maximally commuting with this U(1) generated by $T$? Let us call this as a maximally commuting structure inside $G$ respect to U(1) from $T$.

What is the maximally commuting structure for $G=$SU($N$)? and for $G=$U($N$)? 
What is the maximally commuting structure for $G=$SO($N$)? and for $G=$O($N$)? 
What is the maximally commuting structure for $G=$Sp($N$)? 

Thank you for the helps from experts! 
p.s. Naively, the maximally commuting structure for $G=$SU($N$) should at least contains U(1)$^{N-1}$. But I am not sure what will it be in general.
p.s.2. please also answer the specific examples given above.

REPLY [3 votes]: The examples that interest you are simple groups. I assume the group is simply connected. The maximal commutative subgroup of a semi-simple Lie subgroup is known as the Cartan subgroup. Its algebra is the Cartan subalgebra.
https://en.wikipedia.org/wiki/Semisimple_Lie_algebra#Definition_of_a_Cartan_subalgebra
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